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HANDBOOK 

FOR 

8UKYEYOR8. 


BY 

MANSFIELD  MERRIMAN, 

Professor  of  Civil  Engineering  in  Lehigh  University, 
AND 

JOHN  P.  BROOKS, 

Professor  of  Civil  Engineering  in  State  College  of  Kentucky. 


SECOND  EDITION. 

SECOND  THOUSAND, 


NEW  YORK: 

JOHN  WILEY  & SONS. 

London  : CHAPMAN  & HALL,  Limited, 


, >,{i  pi®  11 
IJR  >.  i*  j 


Copyright,  1895, 

BY 

MANSFIELD  MERRIMAN 

AND 

JOHN  P.  BROOKS. 


ROflERT  DRUMMOND,  ELRCTROTYPER  AND  PRINTER,  NEW  YORK, 


i I IL 


5'^%eWo’r6‘ " 


i!  H''fil 


PREFACE. 


This  work  is  designed  for  the  use  of  classes  in  technical 
schools,  and  also  as  a field  hook  for  surveyors.  It  is  intended 
i ;stt)  embrace  in  concise  form  the  ground  that  a student  should 


cover  in  surveying  before  taking  up  the  subject  of  railroad 
5 lWtinn  Hence  it  includes  the  fundamental  theoretical  prin- 


lo  cation. 

'^ciples,  land  and  town  surveying,  leveling  and  simple  triangula* 
tion,  and  topography.  The  attempt  has  been  made  to  discuss 
each  of  these  topics  clearly  and  concisely,  and  in  accordance 
with  the  best  modern  methods. 

The  need  of  the  volume  arose  merely  from  the  fact  that  no 
text-book  on  elementary  surveying  in  pocket-book  form  could 
be  found  in  the  market.  While  in  the  field  a student  should 
have  a book  of  tables  ever  at  hand,  and  if  these  are  combined 
with  the  text  a double  advantage  is  often  found,  particularly 
in  adjusting  instruments  and  in  arranging  forms  for  notes. 

In  arranging  the  order  of  presentation  the  rule  has  been  as 
far  as  possible  to  proceed  from  the  simple  to  the  complex  in  a 
natural  order.  For  instance,  the  most  difficult  thing  in  sur 
|^eying  is  the  determination  of  a true  meridian,  and  hence  in 
j this  volume  it  comes  last  of  all,  although  in  most  other  books 
y it  is  presented  at  an  early  stage. 

jv.  As  all  persons  likely  to  use  the  volume  have  access  to  sur* 
v veying  instruments,  no  illustrations  of  these  are  given.  The 
i>^effort  has  been  made,  however,  to  set  forth  methods  of  testing 
and  comparing  instruments  more  fully  than  is  usually  done  in 
elementary  books.  As  an  instance  of  this,  attention  is  called 
to  the  determination  of  the  eccentricity  of  the  graduated  circle 
^ v.  of  a transit  given  in  Article  27 . 

The  old  terms  “latitude”  and  “ departure,”  borrowed  from 
1 A navigation,  are  not  here  used,  but  instead  “ latitude  difference” 
fX  and  “longitude  difference”  are  employed,  as  is  universally 

3 

I • )0  1 A*ZH 


/ n r 


4 PREFACE. 

done  in  geodetic  surveying  , the  terms  “ latitude  and  “ longi- 
tude ” are  moreover  used  in  the  same  sense  as  in  geodesy  and 
astronomy  That  this  method  has  advantages  the  experience 
of  many  years  of  teaching  may  hear  witness. 

The  first  field  work  done  by  a student  is  usually  plotted  to 
a large  scale,  and  hence  in  Chapter  IV  the  effort  is  made  to 
clearly  distinguish  between  large-scale  and  small-scale  topog- 
raphy Both  the  transit  and  the  plane-table  method  of  stadia 
work  are  presented,  but  preference  is  given  to  the  former. 
Hydrographic  and  mine  surveying  are  briefly  outlined,  the 
latter  being  with  especial  reference  to  the  practice  in  the  an- 
thracite regions  of  Pennsylvania. 

The  tables  of  natural  functions  are  given  to  five  decimal 
places,  while  logarithms  and  logarithmic  functions  are  given 
to  six  decimals  The  old-fashioned  traverse  table  is  omitted, 
as  it  is  of  no  use  when  sines  and  cosines  are  at  hand.  The 
tables  for  stadia  reductions  are  those  computed  by  Professor 
Arthur  Winslow  for  two-minute  intervals  of  vertical  angles. 
For  assistance  in  compiling  Tables  III,  V,  and  VI,  acknowl 
edgments  are  due  to  the  United  States  Coast  and  Geodetic 
Survey 

Mansfield  Merriman 

NOTE  TO  THE  SECOND  EDITION. 

All  errors  discovered  in  the  first  edition  have  been  corrected. 
The  new  method  of  deteimining  a true  meridian  by  an  altitude 
of  the  sun  taken  with  an  engineer’s  transit  is  explained  in  full 
and  a table  of  mean  refractions  is  given.  A chart  of  lines  of 
equal  magnetic  declination  for  the  year  1900,  reproduced  from 
an  advance  sheet  kindly  furnished  by  the  United  States 
Coast  and  Geodetic  Survey,  accompanies  this  edition. 

July,  1897. 


I 


,i  if;  l!/, l: 

di;n'i  pirFfif 


CONTENTS. 


Chapter  I. 

FUNDAMENTAL  PRINCIPLES. 

ART.  PAGE 

1 . Geometry  and  Trigonometry . — . 7 

2.  Lines,  Angles,  and  Azimuths  . . . 10 

3.  Coordinates;  Latitudes  and  Longitudes 13 

4.  Areas  of  Triangles  and  Trapezoids.  15 

5.  Areas  of  Polygons.. . . . . „ 17 

6.  Computation  of  Areas  . . . ......  20 

7.  Division  of  Land . . . . 23 

8.  Inaccessible  Distances 25 

9.  Elevations  and  Heights 27 

1 0.  Errors  of  Measurements  29 


Chapter  II. 

LAND  SURVEYING. 

11.  Chains  and  Tapes 32 

12.  The  Transit 35 

13.  The  Magnetic  Needle . 40 

14.  Field  Work.  . 44 

15.  Survey  of  a Farm  47 

16.  Office  Work 54 

17.  Random  Lines. ...  57 

18.  Resurveys  59 

19.  Traversing . * 62 

20.  United  States  Public-Land  Surveys 64 


Chapter  III, 

LEVELING  AND  TRIANGULATION. 


21.  The  Level — 67 

* 22.  Adjustments  of  the  Level 68 

23.  Comparison  of  Levels 70 

24.  Leveling 73 

25.  Contours  and  Profiles 75 

26.  Adjustments  of  the  Transit 78 

27.  Comp*»\’is'Mi  of  Transits 81 


5 


V 1J  / 1 1 V 

C CONTENTS. 

!|  •';(!  YNWiYIliil 

ART.  , PACE 

28.  Standard  Tapes. . . . » 84 

29.  Base  Lines „ 87 

30.  Triangulation  Work 90 

Chapter  IVe 

TOPOGRAPHIC  SURVEYING 

31 . Large-Scale  Topography , . , . . t 94 

32.  Small-Scale  Topography , 98 

33  Theory  of  the  Stadia. . 100 

34.  Field  Work  with  the  Stadia  . , 105 

35.  Office  Work  108 

36  The  Plane  Table HO 

37  The  Three-Point  Problem  113 

38.  Hydrographic  Surveying  115 

39.  Mine  Surveying,  119 

40.  The  True  Meridian..,  123 

41.  Azimuth  by  Altitude  of  the  Sun  243 

Tables. 

I.  Natural  Sines  and  Cosines 129 

II.  Natural  Tangents  and  Cotangents 139 

Lengths  of  Circular  Arcs 151 

III.  Daily  Variation  of  the  Magnetic  Needle . 152 

IV.  Degrees  of  Longitude  and  Time  153 

V.  Elongations  and  Culminations  of  Polaris 154 

VI.  Azimuths  of  Polaris  at  Elongation 156 

VII.  Metric  and  English  Measures. 158 

VIII.  Length  of  Arcs  of  Latitude  and  Longitude  159 

IX.  Reduction  of  Inclined  Distances  to  the  Horizontal 160 

X.  Stadia  Reductions  for  Reading  100 161 

XI.  Logarithms  of  Numbers  169 

Constant  Numbers  and  their  Logarithms 196 

XII.  Logarithmic  Sines,  Cosines,  Tangents,  and  Cotangents 197 

XIII.  Mean  Refractions 245 

Plates. 

I.  Isogonic  Chart  of  United  States  for  1900 128 

II.  Alphabets  of  Map  Letters 246 


A HANDBOOK  FOR  SURVEYORS. 


CHAPTER  I. 

FUNDAMENTAL  PRINCIPLES. 

Art.  1.  Geometry  and  Trigonometry. 

Geometry  and  Surveying  were  originally  synonymous,  as 
tlie  etymology  of  tlie  former  word  indicates.  They  originated 
in  Egypt,  where  monuments  and  boundary  lines  were  annu- 
ally obliterated  by  the  inundation  of  the  Nile.  Euclid,  pro- 
fessor of  mathematics  at  Alexandria  about  250  b.c.,  wrote  a 
treatise  on  geometry  which  has  never  been  equaled  in  logical 
methods.  Geometry  furnishes  the  principles  on  which  the 
operations  of  surveying  are  founded,  whereby  line  and  angle 
measurements,  the  computation  of  areas,  and  the  construction 
of  maps  are  effected.  Arithmetic  and  Trigonometry  are  the- 
tools  by  which  the  principles  of  Geometry  are  applied. 

The  following  theorems  of  plane  geometry  are  perhaps 
those  of  greatest  importance,  but  many  others  are  constantly 
used  in  the  field  practice  of  engineers  : 

If  two  straight  lines  intersect,  the  opposite  angles  are  equal. 

Straight  lines  parallel  to  the  same  straight  line  are  parallel 
to  each  other. 

The  sum  of  the  interior  angles  of  a polygon  is  equal  to 
twice  as  many  right  angles  as  the  polygon  has  sides  minus 
four  right  angles. 

The  sum  of  the  exterior  angles  formed  by  producing  the  sides 
of  a polygon  is  equal  to  four  right  angles. 

The  square  upon  the  liypothenuseof  a right-angled  triangle 
‘ is  equal  to  the  sum  of  the  squares  upon  the  other  two  sides. 

Angles  at  the  center  of  a circle  are  in  the  same  ratio  as  their 
intercepted  arcs. 

An  angle  at  the  circumference  of  a circle  is  measured  by  one 
half  the  arc  intercepted  by  its  sides. 


7 


8 


FUNDAMENTAL  PRINCIPLES. 


If  the  angles  of  two  triangles  are  equal  each  to  each,  the 
homologous  sides  are  proportional  and  the  triangles  are 
similar. 

The  areas  of  similar  polygons  are  as  the  squares  of  their  ho- 
mologous sides. 

The  area  of  a triangle  is  measured  by  one  half  the  product 
of  its  base  and  altitude.  The  area  of  a trapezoid  is  measured 
by  one  half  the  product  of  the  sum  of  its  parallel  sides  by  its 
altitude. 

The  area  of  a sector  of  a circle  is  measured  by  one  half  the 
product  of  its  arc  and  radius. 

The  circumference  of  a circle  is  equal  to  its  diameter  mul- 
tiplied by  3.1415927.  The  area  of  a circle  is  equal  to  the 
square  of  its  radius  multiplied  by  3.1415927.  ' 


Trigonometry,  or  the  solution  of  triangles  by  means  of  sines 
and  tangents  of  the  angles,  originated  in  the  thirteenth  cen- 
tury, previous  computations  having  been  made  with  chords. 
The  following  rules  for  the  solution  of  oblique  triangles  are 
here  given  for  reference,  but  it  should  be  remembered  that  no 
surveyor  can  attain  success  unless  he  is  thoroughly  conversant 
with  all  of  them  without  the  necessity  of  referring  to  a book. 

A In  any  triangle  let  a , b,  c,  be 

A the  sides  opposite  the  angles 

I \ A,  By  G.  These  sides  are  pro- 

b \ portional  to  the  sines  of 
/ \ opposite  angles.  The  value  of 

C a q each  side  may  be  expressed  in 
Fig.  1.  three  ways  in  terms  of  the  other 

sides  and  angles;  thus, 

+ 2bc  cos  A ; 

sin  B sin  0 


b = „ !iLf  = c = |/«*  + c’  — 2ac  cos  ”5; 
sin  A sin  <J 


c = 6-^  = Va*  + 62  - 2«&  cos  G. 

sin  A sin  b 


Also  each  angle  may  be  expressed  as  follows  : 


sin  A = 7 sin  B = - sin  (7,  cos  A = 
b c 


52  _|_  c2  _ a* 
2 be  ; 


GEOMETRY  AND  TRIGONOMETRY. 


9 


. „ b . . b . ^ tf  + c'-b* 

sin  B — sin  A = - sin  6,  cos  B = — 4r ; 

a c 2 ac 

. n c . c . a'2  + &2  — c2 

sin  (7  = - sin  = r sin  B,  cos  (7  = ^—r . 

a 6 2 ab 

If  A be  made  a right  angle  these  reduce  to  the  formulas  for 
right  triangles,  which  are  too  well  known  to  he  repeated  here. 


While  the  above  expressions  are  sufficient  for  the  solution  of 
all  plane  triangles,  there  are  other  formulas  more  convenient 
for  logarithmic  computation  for  certain  special  cases.  Tables 
of  natural  functions  are  generally  used  in  ordinary  surveying, 
particularly  in  the  field,  while  logarithmic  tables  are  perhaps 
better  for  rapid  work  in  the  office.  The  young  surveyor 
shpuld  be  prepared  to  solve  triangles  quickly  and  rapidly  b} 
either  method. 


When  two  sides  and  the  included  angle  are  given,  as  a,  b, 
and  (7,  the  sum  of  angles  A and  B is  known,  and  their  differ- 
ence may  be  computed,  if  logarithms  are  used,  from 

tan  i(A  — B)  = a ^tan  \(A  + B ), 

CL  — j—  0 

and  then  A and  B are  determined.  When  natural  functions 
are  used  it  will  often  be  more  advantageous  to  use 
sin  G . ^ sin  G 


tan  A — 


tan  B = 


cos  G 

a 


cos  G 


from  which  A and  B can  be  computed  independently,  and 
their  sum  should  equal  180°  — G. 


When  the  three  sides  by  c , are  given  the  cosines  of  the 

angles  can  be  independently  computed  from  the  formulas 
above  given. ' But  some  prefer  to  divide  the  triangle  into  two 
right-angled  triangles  by  dropping  a perpendicular  from  A 
upon  the  base  af  thus  dividing  it  into  two  segments,  di  and  a2. 
The  sum  of  these  segments  is  a,  their  difference  is 

„ _ _(6  + c)(6-c) 

a 

and  then  the  values  of  ax  and  are  found.  Lastly,  the 
angles  B and  G are  computed  from  cos  B = &2  -f-  c and 
cos  G — cm  -v-  b. 


10 


FUNDAMENTAL  PRINCIPLES. 


In  all  kinds  of  computations  a neat  and  orderly  arrangement 
should  be  followed,  and  it  is  recommended  that  all  problems 
given  in  these  pages,  as  well  as  those  arising  in  field  practice, 
should  be  solved  in  ink  in  a special  book  and  be  preserved  for 
reference.  Check  computations  should  in  all  cases  be  made  ; 
this  can  be  done  by  finding  the  same  quantity  in  different 
ways,  by  computing  the  three  angles  independently  and  taking 
their  sum,  or  by  using  both  natural  functions  and  logarithmic 
tables. 

Prob.  1.  Given  a = 227.52  feet,  b = 108.00  feet,  C=  162°  14' ; 
to  compute  independently  the  angles  A and  B. 

Art.  2.  Lines,  Angles,  and  Aztmuths. 

The  measurement  of  a line  consists  in  finding  how  many 
times  it  contains  the  unit  of  measure.  For  several  centuries 
the  Gunter’s  chain  of  66  feet  has  been  the  English  linear  unit 
for  land  measurements  ; it  is  divided  into  100  parts,  called 
links,  and  lengths  are  expressed  in  chains  aud  links,  the  latter 
being  written  as  decimals  of  a chain  ; thus  12  chains  and  72 
links  is  12.72  chains.  Although  this  chain  is  rapidly  going  out 
of  use,  the  young  surveyor  should  be  acquainted  with  it,  since 
a large  part  of  the  land  records  in  the  United  States  is  based 
upon  it. 

In  computing  areas  the  chain  has  the  advantage  that  square 
chains  are  easily  reduced  to  acres  by  moving  the  decimal  point 
one  place  to  the  left.  This  is  because  66  feet  X 66  feet  = 4356 
square  feet,  which  is  one  tenth  of  an  acre.  For  example,  a rect- 
angular lot  6.48  chains  long  and  2.15  chains  wide  contains 
13.932  square  chains,  or  1.3932  acres. 

The  unit  of  linear  measure  now  generally  used  in  the  United 
States  is  the  foot.  In  measuring  lines  a chain  100  feet  long 
divided  into  100  links,  is  used,  and  distances  are  recorded  in 
feet,  decimals  of  a foot  being  estimated  when  possible.  Tapes 
of  various  kinds,  with  the  foot  divided  decimally,  are  also 
used,  especially  in  cities  where  precise . measurements  are 
necessary. 

Custom  and  civil  laws  have  decided  that  the  length  of  the 


LINES,  ANGLES,  AND  AZIMUTHS. 


11 


boundary  line  of  a field  is  not  the  actual  distance  on  the  sur- 
face of  the  ground,  but  that  it  is  the  projection  of  that  dis- 
tance on  a horizontal  plane  In  like  manner,  the  area  of  a 
field  is  not  the  exposed  superficial  surface,  but  the  projection 
of  that  surface  on  a horizontal  plane.  In  all  land  surveying, 
therefore,  horizontal  distances  are  to  be  measured,  and  from 
these  the  areas  are  to  be  computed. 

The  angle  between  two  boundary  lines  of  a field  is  the 
horizontal  angle  between  their  horizontal  projections.  Angles 
are  measured  by  means  of  a graduated  plate  which  can  be 
leveled  so  as  to  be  brought-into  a horizontal  plane.  Although 
it  is  possible  to  make  complete  surveys  by  means  of  the  chain 
alone,  it  is  much  cheaper  to  make  a number  of  angle  measure- 
ments to  be  used  in  connection  with  a few  measured  linear 
distances. 

The  unit  of  angular  measure  is  the  degree,  or  the  ninetieth 
part  of  a right  angle.  The  degree  is  divided  into  sixty  minutes* 
and  the  minute  into  sixty  seconds.  In  rough  land  surveying 
the  angles  are  measured  to  the  nearest  quarter  degree,  in 
ordinary  work  to  the  nearest  minute,  and  in  triangulation  they 
are  expressed  in  seconds. 

An  arc  of  a circle  containing  57.3  degrees,  or  more  accurately 
57.29578  degrees,  is  equal  in  length  to  the  radius.  At  a dis- 
tance of  1000  feet  an  angle  of  one  degree  subtends  an  arc  of 
17.453  feet,  while  an  angle  of  one  minute  subtends  0.291  feet. 
The  sine  of  one  degree  is  0.017452,  and  the  sine  of  one  minute 
is  0.000291.  Thus  for  angles 
less  than  one  degree  the  sub- 
tended arcs  mi  ay  be  taken  as 
closely  proportional  to  their  sines. 

The  angle  which  a line  makes 
with  a standard  line  of  refer- 
ence is  called  the  azimuth  of  the 
line.  The  standard  line  is  usu- 
usually  a north  and  south  line,  or 
meridian.  In  land  surveying 
azimuths  are  measured  from  the  north  around  through  the  east, 


12 


FUNDAMENTAL  PRINCIPLES. 


south  and  west  in  the  direction  of  motion  of  the  hands  of  a 
clock.  Thus  the  azimuth  of  the  north  point  is  0°,  of  the  east 
90°,  of  the  south  180°,  and  of  the  west  270°.  In  Fig.  2 the  azi- 
muth of  the  line  AB  is  60°,  the  azimuth  of  AG  is  150°,  the  azi- 
muth of  AD  is  250°,  and  the  azimuth  if  All  is  290°.  When 
the  azimuths  of  two  lines  are  known,  the  angle  between  them 
is  found  by  taking  the  difference  of  the  azimuths  ; thus  DA11 
rr  290°  - 250°  = 40°. 

The  back  azimuth  of  a line  is  its  azimuth  measured  at  the 
other  end  with  reference  to  a meridian  drawn  through  that 
end.  In  plane  surveying  all  the  meridians  are  parallel,  c;nd 
hence  the  back  azimuth  of  a line  differs  by  180°  from  the  azi- 
muth. For  instance  ih  Fig.  3 let 
the  azimuth  of  AB  be  45°,  then  the 
back  azimuth  is  225°.  In  any  case 
the  back  azimuth  of  a line  BA  is 
the  azimuth  of  AB,  the  initial  let- 
ter indicating  the  end  where  the 
azimuth  is  measured.  In  geodetic 
surveying  the  meridians  converge 
toward  the  pole,  and  hence  the 
back  azimuth  of  a line  differs  from 
its  azimuth  by  an  amount  slightly  greater  or  less  than  180°; 
also  the  south  is  taken  as  the  initial  point,  and  the  azimuths 
are  measured  around  through  the  west,  north,  and  east. 

When  the  interior  angles  of  a polygon  have  been  measured 
and  also  the  azimuth  of  one  of  its  sides,  the  azimuths  of  the 
other  sides  are  easily  found.  No  special  rules  need  be  given 
for  finding  these,  for  no  error  can  occur  if  a sketch  be  drawn 
in  each  particular  case.  For  example,  in  Fig.  3,  if  the  angle 
B is  75°  and  the  azimuth  of  AB  is  45°,  then  the  azimuth  of 
BC  is  150°  ; if  further  the  angle  G is  40°,  then  the  azimuth 
of  GD  is  290°,  and  so  on. 

Prob.  2.  A polygon  of  six  sides  has  the  interior  angles  A 
- 58°  24' , B = 121°  30',  G = 123°  30',  D = 188°  15',  E = 95° 
19',  F = 133°  02',  and  the  azimuth  of  AB  is  0°  00'.  Find  the 
azimuth  of  each  of  the  other  sides. 


LATITUDES  AND  LONGITUDES. 


13 


Aiit.  3.  Latitudes  and  Longitudes. 


In  geography  tlie  latitude  of  a point  is  its  angular  distance 
north  or  south  from  the  equator,  and  the  longitude  of  a point 
is  its  angular  distance  west  or  east  from  an  assumed  meridian. 
In  plane  surveying  the  meanings  of  the  words  are  analogous, 


C 

/ I 

ci\ — j^A 


but  the  distances  are  measured  in  feet  from  any  two  conven- 
ient lines  of  reference  which  intersect  at  right  angles;  one  of 
these  lines  is  generally  a north  and  south  line  or  meridian. 
Thus  in  Fig.  4 let  SN  be  a meridian  N 

and  WE  be  a line  perpendicular  to  it. 

Let  A and  B be  the  ends  of  the  line 
AB,  and  from  each  let  perpendiculars 
be  drawn  to  NS  and  WE.  Then  A 
and  biB  are  the  latitudes,  and  a A and 
bB  are  the  longitudes  of  the  points  A 
and  B.  Latitudes  of  points  north  of 
WE  are  regarded  as  positive,  while 
those  of  points  south  of  it  are  negative.  Longitudes  east  of 
NS  are  positive,  while  those  west  of  NS  are  negative.  Thus 
the  point  G has  a positive  latitude  and  a negative  longitude. 


w o|  aT  * 

v I 
I 


G 

$ 

Fig.  4. 


The  difference  of  the  latitudes  of  the  ends  of  a line  is  called 
the  latitude  difference  of  that  line;  thus  ab  is  the  latitude 
difference  of  AB.  The  difference  of  the  longitudes  of  the 
ends  of  a line  is  called  the  longitude  difference  of  that  line  : 
thus  ctibi  is  the  longitude  difference  of  AB.  In  general  let 
Li  and  X2  be  the  latitudes  of  two  points,  and  Mi  and  M*  their 
longitudes;  then  Z^— Z2  is  the  latitude  difference  and  Mx  — M 2 
is  the  longitude  difference. 

When  the  'length  and  azimuth  of  a line  are  known  its  lati- 
tude and  longitude  differences  are  found  by  multiplying  the 
length  by  the  cosine  and  sine  of  the  azimuth.  Thus,  from 
Fig.  4, 

Latitude  difference  of  AB  — ab  — l cos  Z. 

Longitude  difference  of  AB  = aibx  —l  sin  Z. 

For  example,  let  the  length  of  a line  be  457.69  feet  and  its 
azimuth  be  279°  OF  44";  then  its  latitude  difference  is  + 71.83 
feet  and  its  longitude  difference  is  — 452  02  feet- 


14 


FUNDAMENTAL  PKINCIPLES. 


When  the  latitude  Li  and  longitude  Mi  of  a point  are  known, 
as  also  the  length  and  azimuth  of  a line  joining  that  point 
with  another,  the  latitude  Z2  and  the  longitude  M2  of  the 
second  point  are 

Li  = Li  -f-  l cos  Z , = Mi  + l sin  Z . 

The  proof  of  these  equations  is  readily  seen  from  Fig.  4, 
taking  A as  the  first  point  and  B as  the  second. 


The  latitude  and  longitude  of  a line  are  often  called  coor- 
dinates, while  the  two  standard  reference  lines  SN  and  WE 
are  called  the  coordinate  axes,  and  their  intersection  0 is  known 
as  the  origin  of  coordinates.  The  latitudes  and  longitudes  of 
points  in  the  four  quadrants  formed  by  these  axes  have  the 
same  signs  as  sines  and  cosines  in  trigonometry.  It  is  usual  in 
land  surveys  to  assume  the  coordinate  axes  in  such  positions 
that  all  the  points  of  the  survey  will  fall  in  the  NE  quadrant 
where  their  latitudes  and  longitudes  are  positive.  Thus  Fig. 
5 shows  a field  ABCD  with  the  coordinates  of  each  corner 
positive  with  respect  to  the  two  axes. 


A line  whose  azimuth  is  known,  is  often  called  a course,  the 
word  course  implying  a definite  direction.  Lines  or  courses 
running  northward,  or  toward  the 
top  of  the  page,  are  called  north 
courses,  while  those  that  run  south- 
ward are  south  courses ; thus  in  Fig. 
5 the  lines  DA  and  AB  are  north 
courses,  while  BG  and  CD  are  south 
courses.  Lines  running  eastward, 
or  toward  the  right  of  the  page,  are 
called  east  courses,  whiie  those  run- 
ning westward  are  west  courses; 
thus  AB  and  BG  are  east  courses,  while  CD  and  DA  are  west 
courses. 

The  latitude  difference  of  a north  course  is  positive  and  is 
called  a northing,  while  that  of  a south  course  is  negative  and  is 
called  a southing;  thus  ab  is  positive,  but  be  is  negative.  The 
longitude  difference  of  an  east  course  is  positive  and  is  called  an 
easting,  while  that  of  a west  course  is  negative  and  is  called  a 


AREAS  OF  TklANGLES  ANI)  TRAPEZOIDS.  15 


westing;  thus  biCi  is  positive,  but  Cidi  is  negative.  If  atten- 
tion be  paid  to  the  signs  of  the  cosines  and  sines  of  the  azi- 
muth in  making  the  computations,  the  latitude  and  longitude 
differences  will  always  come  out  with  their  proper  signs.  In 
many  books  on  surveying  the  northings  and  southings  are 
called  latitudes  instead  of  latitude  differences,  while  the  east- 
ings and  westings  are  called  departures  instead  of  longitude 
differences  ; but  the  plan  here  adopted  is  more  in  accordance 
with  the  methods  of  geodesy. 

Prob.  3.  Given  the  latitude  of  one  end  of  a line,  as  -f-  2804.4, 
its  longitude  as -f  4661.3,  its  length  797.2  feet,  and  its  azimuth 
115°  44'  28".  Compute  the  latitude  and  longitude  of  the  other 
end.  (Draw  a figure  before  beginning  the  solution.) 

Art.  4.  Areas  of  Triangles  and  Trapezoids. 

The  areas  of  fields  are  usually  expressed  in  acres,  square 
rods,  and  square  feet,  there  being  160  square  rods  in  an  acre 
and  2721  square  feet  in  a square  rod.  In  rough  land  surveys 
the  area  is  expressed  in  acres,  roods,  and  square  rods,  a rood 
being  one  fourth  of  an  acre.  In  speaking  of  areas  a square 
rod  is  usually  called  simply  a rod. 

The  area  of  any  triangle  is  equal  to  one-half  the  product  of 
the  two  sides  into  the  sine  of  their  included  angle.  Thus,  if 
a , b,  c,  be  the  sides  opposite  the  angles  A,  B,  (7,  respectively, 
the  area  can  be  expressed  in  three  ways, 

Area  — % ab  sin  C = l ac  sin  B = %bc  sin  A; 

and  if  one  of  the  angles,  as  A,  is  a right  angle,  the  area  is 
simply  \bc.  As  an  example,  let  a = 22.00  chains,  c = 13.20 
chains,  and  B = 53°  08' ; from  Table  I sin  B & found  to  be 
0.80003,  and  then  the  area  is  11.616  square  chains,  or  1 acre, 
25  square  rods  and  233  square  feet.  . 

When  the  three  sides  of  a triangle  have  been  measured  its 
area  may  be  found  by  the  following  rule  : Add  together  the 
three  sides  and  take  half  their  sum,  from  the  half-sum  sub- 
tract each  side  separately,  multiply  together  the  half-sum  and 
the  three  remainders,  and  take  the  square  root  of  the  product. 


16 


FUNDAMENTAL  PRINCIPLES. 


Or,  let  a,  b}  c,  be  the  three  sides,  and  s the  half -sum  % (a  -J-  b 
+ c)\  then 

Area  = \^s(s— a)(s— b)(s— c). 

For  example,  let  a = 220  feet,  b = 176  feet,  and  c = 132  feet ; 
then  s = 264,  s—a  = 44,  s— b — 88,  s—c  — 132,  and  the  area  is 
11616  square  feet,  or  42  J square  rods. 

If  the  latitudes  and  longitudes  of  the  vertices  of  a triangle 
with  respect  to  a meridian  ON  and  a parallel  OE  are  given, 
the  area  of  the  triangle  is  easily 
computed,  it  being  the  difference 
between  the  area  of  a rectangle 
and  of  three  right-angled  tri- 
angles. For  example,  let  the 
latitudes  of  the  points  A,  By  and 
ol  E G in  Fig.  6 be  400,  350,  and  100 

Fig-  6-  feet  respectively,  and  the  corre- 

sponding longitudes  be  500,  700,  and  80  feet.  Then  the  height 
of  the  rectangle  is  300  feet  and  its  width  is  620  feet,  which 
give  186,000  square  feet  for  its  area.  The  sum  of  the  areas 
of  the  three  right-angled  triangles  is  124,500  square  feet. 
Hence  the  area  of  A B G is  1 acre  and  17,940  square  feet. 

The  area  of  a trapezoid  is  equal  to  half  the  sum  of  the  par- 
allel sides  multiplied  by  its  altitude.  The  trapezoids  of  most 
common  occurrence  in  surveying  have  two  right  angles,  as  for 
instance  aABb  in  Fig.  5,  whose  area  is  \(aA  -)-  bB)ab.  In 
order  to  determine  the  area  of  an  irregular  figure  like  that  of 
ABCJD  in  Fig.  7,  perpendiculars,  or  offsets,  are  sometimes 
erected  upon  the  straight  line  AD  and  their  lengths  measured 
as  well  as  their  distances  apart,  the  distances  be , cd , etc.,  being 


such  that  Bcu  c^dXy  etc.,  may  be  regarded  as  practically 
straight.  Then  the  total  area  is  the  sum  of  the  areas  of  the 


AREAS  OF  POLYGONS. 


17 


triangle  ABb,  and  of  tlie  trapezoids  bBciC,  ccxd\dy  etc.  This 
method  is  particularly  applicable  to  cases  where  the  lengths  of 
the  offsets  are  less  than  one  or  two  chains  and  where  great 
precision  is  not  required. 


The  area  of  any  polygon  may  be  determined  by  dividing  it 
into  triangles.  Fig.  8 shows  two 
ways  of  thus  dividing  a six-sided 
field,  and  many  others  are  pos- 
sible. In  practice  it  is  more  ad-  C 
vantageous  to  measure  a number 
of  angles  and  a few  sides,  rather 
than  all  the  sides  of  all  the  tri- 
angles. But  a better  method  for  computing  the  area  of  a 
- polygon  is  by  means  of  trapezoids,  as  explained  in  the  next 
article. 


Prob.  4.  Compute  the  area  of  the  first  diagram  in  Fig.  8 
from  the  following  data  : AB  — 317.8  feet,  BF  — 284.3  feet, 
FA  = 250.5  feet,  FG  = 512  7 feet,  FD  = 510.0  feet,  DEF  = 
90°  00',  EFD  = 69°  45',  DFG  = 61°  12',  CFB  = 49°  30'. 


Art.  5.  Areas  of  Polygons. 


To  determine  the  area  of  a polygonal  field  it  is  customary  to 
measure  the  length  of  each  side  and  each  of  the  interior  angles. 
The  azimuth  of  one  side  is  also  either  determined  or  assumed  ; 
then  by  Art.  2 the  azimuth  of  each  of  the  other  sides  is  readily 
found.  Let  ABODE  A in  Fig.  9 be  a field  in  which  the  length 
and  azimuth  of  each  side  is  known. 

It  is  required 'to  deduce  a method 
for  computing  the  area. 

Let  a meridian  be  drawn  through 
the  most  westerly  corner  of  the 
field,  and  from  each  of  the  other 
corners  let  perpendiculars  Bb,  Gc , 

Dd,  and  Ee  be  drawn  to  it ; these 
are  the  longitudes  of  the  corners 
(Art.  3).  Then  the  area  of  the 
field  is  equal  to  the  area  bBCDd  minus  the  areas  AbB  and 


18  FUNDAMENTAL  PRINCIPLES. 

AEDd.  The  first  area  is  formed  by  tbe  two  trapezoids  bBCc 
and  cCDd,  tbe  second  is  tbe  triangle  AbB,  while  tbe  third  is 
formed  by  tbe  triangle  AEe  and  tbe  trapezoid  eEDd.  Hence 
Area  = i(bB cG)bc i(cG+ dB)cd 

— ibB . Ab  — \eE.  eA  — %(dD  -f-  eE)de , 
and  tbe  double  area  of  tbe  field  is 

2 Area-  = (bB+  cO)bc+  (cG  + dD)cd  - bB . Ab 

— eE . eA  — (dJD  -)-  eE  )de , 

and  it  has  been  shown  in  Art.  3 bow  all  tbe  quantities  in  this 
expression  can  be  computed. 

Tbe  longitude  of  a point  is  its  distance  from  tbe  meridian 
(Art.  3);  thus  bB  and  cC  are  the  longitudes  of  tbe  points  B and 
G.  Tbe  longitude  of  a line  or  course  may  now  be  defined  to 
be  tbe  longitude  of  its  middle  point,  thus  %{bB-\-  cG)  is  tbe 
longitude  of  tbe  course  BG.  Hence  bB-\-cG  is  tbe  double 
longitude  of  BG,  or  tbe  double  longitude  of  any  course  is  tbe 
sum  of  tbe  longitudes  of  its  ends. 

Inspection  of  tbe  above  expression  for  tbe  double  area  of  a 
field  shows  two  facts  : First,  that  tbe  double  area  is  tbe  differ- 
ence of  two  quantities,  one  being  tbe  sum  of  tbe  areas  of  tbe 
trapezoids  included  between  tbe  south  courses  and  tbe  meridian, 
while  tbe  other  is  the  sum  of  tbe  areas  of  tbe  trapezoids  and 
triangles  included  between  tbe  north  courses  and  tbe  meridian. 
Second,  that  each  of  these  areas  is  tbe  product  of  tbe  double 
longitude  of  a course  by  its  latitude  difference.  Hence  let 
Si  , $2 , etc. , be  tbe  double  longitudes  of  tbe  south  courses  and 
Si  , s2 , etc.,  their  southings,  and  let  ATi  , AT2 , etc.,  be  tbe 
double  longitudes  of  tbe  north  courses,  and  rh  , n 2 , etc.,  their 
northings  ; then 

2 Area  — SiSi  — |—  S^s^  — |—  etc.  — Nifii  — Nz'fi'z  etc. 
gives  a general  rule  for  computing  tbe  area  of  any  polygonal 
field.  Tbe  areas  SiSi , $2s2 , etc.,  are  often  called  south  areas, 
while  tbe  others  are  called  north  areas. 

Tbe  northings  and  southings  of  each  course  having  been 
computed  by  Art.  3,  as  also  tbe  eastings  and  westings,  it  only 
remains  to  find  tbe  double  longitudes.  For  tbe  first  course 


AREAS  OF  POLYGONS. 


19 


AB  the  double  longitude  is  its  easting  bB.  For  the  second 
course  BG  the  double  longitude  is  bB- f-  cG,  that  is,  bB  -f  bB  -f* 
bi  G.  For  the  third  course  GD  the  double  longitude  is  c G + dD , 
that  is,  bB cG -\-  biG  — Gdi.  In  general  the  following  rule 
will  be  useful : 

The  double  longitude  of  any  course  is  equal  to  the  double 
longitude  of  the  preceding  course  plus  the  longitude  differ- 
ence of  that  course  plus  the  longitude  difference  of  the 
course  itself. 

When  the  longitude  difference  is  negative,  or  a westing,  it  is 
used  with  the  minus  sign  and  hence  subtracted  instead  of 
added.  If  the  meridian  is  drawn  through  the  most  westerly 
corner  of  the  field,  as  in  Fig.  9,  all  the  double  longitudes  are 
positive.  As  a check  on  the  work  the  double  longitude  of  the 
last  course  will  be  found  equal  to  its  westing  ; thus  the  double 
longitude  of  EA  is  eE. 

The  following  steps  in  the  computation  of  the  area  of  a po- 
lygonal field  may  now  be  enumerated  : 

1st.  Measure  the  length  of  each  side  or  course  and  each  of 
the  interior  angles  ; these  constitute  the  field  notes.  Also 
measure  the  azimuth  of  one  of  the  courses,  or  if  this  is  not 
measured  assume  any  value  for  this  azimuth. 

2d.  Compute  the  azimuth  of  each  of  the  other  courses  (Art.  2). 
3d.  Compute  the  latitude  difference  and  the  longitude  differ- 
ence for  each  course  (Art.  3). 

4tli.  Compute  the  double  longitude  for  each  course. 

5th.  Multiply  each  double  longitude  by  its  latitude  differ- 
ence ; call  the  positive  products  north  areas,  and  the  negative 
products  south  areas. 

6th.  Take  the  sum  of  the  south  areas  and  the  sum  of  the  north 
areas  ; one  half  of  their  difference  will  be  the  area  of  the  field. 

In  Art.  6 a numerical  example  will  be  given  illustrating  the 
computations  in  full. 

' Prob.  5.  A triangle  ABG  has  sides  with  the  following 
lengths  and  azimuths  : 

AB,  l = 312.0  feet,  z ==  45  degrees. 

BG,  l — 540.4  feet,  z — 135  degrees. 

GA,  l = 624.0  feet,  z = 285  degrees. 

Compute  the  latitude  differences,  the  longitude  differences 
and  the  double  longitudes  for  each  course. 


20 


FUNDAMENTAL  PRINCIPLES. 


Art.  6.  Computation  of  Areas. 


The  following  are  the  lengths  of  the  sides  and  the  interior 
angles  of  a polygon  as  measured  in  laying  out  a field: 

AB  = 800.0  feet,  A = 58°  14' 

BG  = 500.0  feet,  B = 120  00 

CD  = 200.0  feet,  G = 125  00 

DE  = 100.0  feet,  Z)  = 200  00 

EF  = 600.0  feet,  E = 88  84 

JK4  = 700.0  feet,  F = 133  12 

No  azimuth  was  taken  in  the  field,  and  hence  for  the  pur- 
pose of  computing  the  area  the  meridian  is  assumed  to  pass 
through  AB,  so  that  the  azimuth 
of  AB  is  0°  00'. 

The  first  step  is  to  find  the  azi- 
muths of  the  other  sides  by  the 
method  of  Art.  3.  In  general  the 
azimuth  of  any  course  is  equal  to 
that  of  the  preceding  course,  plus 
180  degrees,  minus  the  interior  an- 
gle between  the  two  courses.  Thus 
the  azimuth  of  BG  is  0°  -|-  180°  — 
120°  = 60°;  the  azimuth  of  GD  is 
60°  -j-  180°  — 125°  = 115°,  and  so  on. 
As  a check  on  the  work  the  azimuth  of  AB  computed  from 
that  of  FA,  should  be  found  to  be  0°  00'. 


The  latitude  and  longitude  differences  of  the  courses  are  next 
computed  as  follows,  by  Art.  3 : 

Lat.  Diff.  AB  = 800  cos  0°  00'  ==  + 800.00 
Lat.  Diff.  BG  = 500  cos  60°  00'  = + 250.00 
Lat.  Diff.  GD  = 200  cos  115°  00'  = - 84.52 
Long. Diff.  AB=  800  sin  0°  00'  = 0.00 

Long.  Diff.  BG=  500  sin  60°  00'  = + 433.01 
Long.Diff.  EF=  600  sin  191°  26'  = - 118.94 

In  like  manner  all  the  latitude  and  longitude  differences  are 
computed  and  the  results  are  tabulated,  the  positive  latitude 
differences  being  northings  and  the  negative  ones  southings, 


COMPUTATIOX  OF  AREAS. 


21 


while  the  positive  longitude  differences  are  eastings,  and  the 
negative  ones  westings. 


Lengths, 

feet. 

Lat.  Differences. 

Long.  Differences. 

Courses. 

Azimuths. 

North- 

ings. 

South- 

ings. 

Eastings. 

West- 

ings. 

AB 

800.00 

0°  00' 

800.00 

0.00 

0.00 

BC 

500.00 

60  00 

250.00 

433.01 

CD 

200.00 

115  00 

84.52 

181.26 

DE 

100.00 

95  00 

8.71 

99.62 

EF 

600.00 

191  26 

588.09 

118.94 

FA 

700.00 

238  14 

368.52 

595.14 

Totals 

1050.00 

1019.84 

713.89 

714.08 

Errors 

0.16 

0.19 

Since  the  survey  was  made  by  a circuit  from  A back  to  A it 
is  evident  that  the  sum  of  the  northings  should  equal  the  sum 
of  the  southings  ; also  the  sum  of  the  eastings  should  equal 
the  sum  of  the  westings.  In  practice  this  is  rarely  attained, 
but  there  is  an  error,  called  the  error  of  closure,  which  should 
be  adjusted  before  the  double  longitudes  are  computed.  In 
this  case  the  significance  of  the  errors,  0.16  feet  in  latitude  and 
0.19  feet  in  longitude  is  that,  if  starting  from  A,  the  corners 
were  to  be  accurately  located  from  the  above  data,  the  end  A' 
of  the  line  FA'  would  fall  0.16  feet  to  the  north  of  A'  and  0.19 
feet  west  of  it. 

The  error  of  closure  is  caused  by  errors  in  the  measurement 
of  the  lines,  or  in  observing  the  angles,  or  in  both.  However, 
if  the  sum  of  the  interior  angles  of  the  polygon  equals  180° 
into  the  number  of  sides  minus  860°,  the  probability  is  that 
the  error  of  closure  is  mostly  due  to  the  linear  measures.  As 
the  error  in  measuring  a line  increases  with  its  length,  the 
error  in  latitude  should  be  distributed  among  all  the  latitude 
differences  in  proportion  to  their  lengths,  one  half  of  it  being 
applied  to  the  northings  and  one  half  to  the  southings.  The 
error  in  longitude  is  treated  in  the  same  way.  Thus  in  this 
case  the  errors  per  foot  in  latitude  and  longitude  are 


0.08 

1050 


0.000076  ; 


0.095 

714 


= 0.000138, 


and  the  adjusted  latitude  and  longitude  differences  are  found 
as  follows  : 


FUNDAMENTAL  PRINCIPLES. 


Northing  AB  — 800.00  - 0.000076  x 800  = 799.94 

Southing  CD  = 84.52  + 0.000076  X 84  = 84.53 

Easting  BC  = 433.01  + 0.000133  X 433  = 433.07 

Westing  EF  - 118.94  - 0.000133  X 119  = 118.93 

and  their  values  are  inserted  in  the  table  given  below. 

The  double  longitudes  of  the  courses  are  next  computed. 
For  the  course  AB,  the  double  longitude  is  its  departure  0.00, 
fofthe  second  course  BC  it  is  433.07,  for  CD  it  is  433.07  + 
433.07  + 181.29  = 1047.43,  and  so  on.  As  a check  on  the 
work  the  double  longitude  of  the  last  course  will  be  found 
equal  to  its  westing.  The  fifth  column  of  the  table  gives  all 
i the  double  longitudes. 


Courses. 

Adjusted 
Lat.  Differences 

Adjusted 
Long.  Differences 

Double 

Longi- 

tudes. 

Double 

Areas. 

N. 

S. 

E. 

W. 

North. 

South  . 

AB 

799.94 

0.00 

0.00 

0.00 

0 

BC 

249.98 

433.07 

433.07 

108258.8 

CD 

84.53 

181.29 

1047.43 

88539.3 

DE 

8.71 

99.63 

1328.35 

11569.9 

EF 

588.13 

118.93 

1309.05 

169891 .6 

FA 

368.55 

595.06 

595.06 

219309.4 

1049.92 

1049.92 

713.99 

713.99 

108258.8 

1089310.2 

The  fifth  step  is  to  multiply  the  double  longitude  of  each 
course  by  its  adjusted  latitude  difference,  and  to  place  the 
products  in  the  columns  of  double  areas.  Lastly  each  of  these 
columns  is  added,  and  then  the  double  area  of  the  field  is 
1089  310.2  - 108  258.8  = 981  051.4  square  feet, 
and  accordingly  the  required  area  is  490  525.7  square  feet, 
which  is  equal  to  11  acres,  41  rods,  and  204  square  feet. 

This  result  can  be  verified  by  making  another  computation  in 
which  the  meridian  is  assumed  to  pass  through  some  other 
side,  as  BC.  Then  the  azimuth  of  BC  will  be  0°00',  that  of  CD 
will  be  55°  00'  and  so  on.  A new  set  of  latitude  and  longitude 
projections  is  computed  and  these  are  adjusted  in  the  man- 
ner explained.  The  double  longitudes  of  the  courses  are  then 
found  and  each  is  multiplied  by  its  corresponding  northing 
or  southing.  Lastly  one  half  of  the  difference  of  these  pro- 
ducts will  give  the  area  in  square  feet,  which  should  closely 
agree  with  the  result  found  above. 


DIVISION  OF  LAND. 


23 


Prob.  6.  Compute  the  area  of  the  above  field  taking  the  azi- 
muth of  BG  as  0°  00';  also  taking  the  azimuth  of  EF  as  0 00'; 
also  taking  the  azimuth  of  AB  as  90°  00'„ 

Art.  7.  Division  of  Land. 

An  infinite  number  of  problems  may  arise  in  the  division  of 
a field.  The  simpler  ones  will  be  readily  solved  by  the  use  of 
the  principles  of  geometry.  The  more  difficult  ones  can  be 
solved  after  a complete  survey  of  the  field  and  the  computation 
of  its  area  has  been  made. 

The  first  problem  to  be  considered  is  that  of  dividing  a field 
into  two  given  parts  by  a line  starting  from  a given  point.  As 
an  example  let  the  field  whose  area  was 
computed  in  Art.  6 be  taken,  and  let  it 
be  required  to  draw  from  the  point  D , 
a line  DP  so  that  the  area  BCDP  shall 
be  5 acres,  or  217  800  square  feet.  The 
solution  of  the  problem  involves  the 
determination  of  the  distance  AP  or 
BP , and  of  the  length  and  azimuth  of 
the  dividing  line  DP.  (Fig.  11.) 

Let  a line  be  drawn  from  D to  the 
corner  A,  and  suppose  that  the  area 
ABCDA  can  be  found.  Then  the  area 
of  the  triangle  APDA  is  known,  as  this  is  equal  to  ABCDA 
minus  5 acres.  The  longitude  dD  of  the  point  D is  also 
known.  Hence  the  length  of  AP  is 

* , 2 area  of  APDA 

~ dD  ’ 

and  then  PB  — AB  — AP.  The  length  and  azimuth  of  DP * 
are  finally  computed  from  the  right  triangle  of  dDP. 

To  perform  the  computations  for  finding  the  area  ABCDA , 
the  adjusted  latitude  and  longitude  differences  of  the  courses 
from  A to  D are  to  be  taken  from  Art.  6 and  inserted  in  the 
new  table  given  below.  The  latitude  difference  of  the  course* 
DA  is  then  found  from  the  principle  that  the  sum  of  the  north- 
ings.must  equal  the  sum  of  the  southings,  and  the  longitude 


C 


24 


FUNDAMENTAL  PRINCIPLES. 


Courses. 

Latitude 

Differences. 

Longitude 

Differences. 

Double 

Longi- 

tudes. 

Double  Areas. 

N. 

S. 

E. 

W. 

North. 

South. 

AB 

799.94 

0.00 

0.00 

0.00 

0 

BC 

249.98 

433.07 

433.07 

108258.8 

CD 

84.53 

181.29 

1047.43 

88539.3 

DA 

(965.39) 

(614.36) 

614.36 

593097.0 

1049.92 

(1049.92) 

614.36 

(614.36) 

108258.8 

681636.3 

difference  of  DA  is  supplied  in  like  manner.  Completing  then 
tlie  computations,  the  area  ABGDA  is  found  to  be  286688.7 
square  feet.  The  area  of  the  triangle  ADP  is  this  quantity 
minus  217  800  square  feet,  and  the  distance  AP  is 


AP  = 


2 X 68888.7 
614.86 


= 224.26  feet; 


whence  PB  is  575.68  feet,  and  hence  the  point  P can  be  lo- 
cated from  either  A or  B.  The  azimuth  of  PD  is  determined 
thus, 


tan  dPD  = 


614.86 


Pd  ~ 575.68  + 249.98  - 84.58’ 
from  which  the  angle  dPD  is  found  to  be  89°  89'  26",  which 


is  the  azimuth  of  PD. 


Lastly  the  length  of  PD  is 

PD  = 4^=962.65  feet, 
sin  Z 


and  thus  the  field  is  divided  by  the  line 
DP  so  that  the  area  BCDP  is  5 acres. 

A second  problem  is  that  of  dividing  a 
field  into  two  parts  by  a line  having  a 
given  direction.  ‘ For  example,  let  it  be 
required  to  divide  the  field  ABCDEF  into 
two  parts  by  a line  PQ  so  that  the  azimuth 
of  PQ  shall  be  45  degrees  and  the  area 
PBGDQ  shall  be  5 acres  (Fig.  12).  First, 
the  computation  of  the  entire  field  is  to  be  made  as  in  Art.  6. 
Secondly,  a line  DM  is  drawn  from  the  corner  D,  parallel  to 
QP , and  by  the  method  above  described  the  area  MBGDM  is 
found  to  be  178859.8  square  feet  and  the  length  of  DM  to  be 


Fig.  12. 


INACCESSIBLE  DISTANCES. 


25 


868.84  feet.  The  area  of  the  trapezoid  PMDQ  is  hence  to  be 
88940.7  square  feet.  Let  x be  the  altitude  of  this  trapezoid;  its 
area  is  1>(MD  + PQ)X*  But  PQ  — MD  + x cot  QPM  -f-  x cot 
DQP.  Hence 

£(2 MB  + x cot  QPM+  x cot  JDQP)x  = 38940.7. 

Since  QPM  = 45°  and  DQP  = 50°,  this  reduces  to 
x 2 + 944.8505  = 42347.5, 
from  which  x is  found  to  be  42.87  feet.  Then 

MP  = 42.87  sin  45°  = 60.63  feet, 

DQ  = 42.87  sin  50°  = 55.97  feet, 

PQ  = 868.84  + 42.87  X 1.8391  |S  947.68  feet, 

and  lastly  the  distance  AP  is  found  to  be  290.4  feet.  Thus 
P and  Q are  located  so  that  PQ  has  the  azimuth  45°,  and  the 
area  PBCDQP  is  5 acres.  This  computation  may  now  be 
checked  by  computing  the  area  of  APQEFA , which  should  be 
found  to  be  272725.7  square  feet. 

Prob.  7.  Divide  the  field  ABGDEFA  into  two  equal  parts 
by  a line  PQ  drawn  from  the  middle  point  of  AB.  Also  divide 
it  into  two  equal  parts  by  a line  PQ  drawn  perpendicular  to 
the  side  AB. 

Art.  8.  Inaccessible  Distances. 

A common  problem  in  surveying  is  to  find  the  horizontal  dis- 
tance between  two  points  when  one  or  both  of  them  are  in- 
accessible. This  can  be  solved  in 
many  ways  by  the  application  of  the 
principles  of  geometry  and  trigo- 
nometry. 

In  Fig.  13  let  A be  an  accessible 
point  and  X an  inaccessible  point  on 
the  other  side  of  a river.  It  is  re- 
quired to  find  the  distance  AX  by 
means  of  the  chain  alone.  Place  a 
point  D at  any  convenient  position 
in  the  prolongation  of  XA,  lay  off  a 
distance  AB,  make  BG  equal  to  AD,  Fig.  13. 

and  DC  equal  to  AB,  thus  forming  a parallelogram  ABCD . 


26 


FUNDAMENTAL  PRINCIPLES. 


Mark  a point  E where  XC  cuts  AB , measure  AE,  EB,  and  BC. 
Then  from  the  similar  triangles  CBE  and  EXA , 


AX  = 


AExBG 
BE  9 


by  which  the  required  distance  can  be  computed. 


By  the  use  of  an  instrument  for  measuring  angles  the  field 
operations  become  much  simpler,  and  indeed  the  method  by  the 
chain  is  often  impracticable  when  AX  is  a long  line.  Let  (in 
Fig.  13)  a line  AEbe  measured,  and  also  the  two  angles  A and 
E ; then  the  angle  X is  180°  — A — E,  and 


AX  = AE 


sin  E 
sin  X 9 


which  is  the  required  distance.  The  base  line  AE  should 
usually  be  nearly  as  long  as  the  distance  AX  in  order  to  secure 
the  most  accurate  result,  and  it  is  also  well  that  the  angles  A * 
and  E should  be  approximately  equal. 

Y Tlie  problem  of  two  inaccessi- 

ble points  is  illustrated  in  Fig.  14. 
Here  the  distance  XFis  required, 
and  for  this  purpose  a base  line 
AB  is  measured  in  a convenient 
location,  and  as  nearly  parallel 
to  XY  as  practicable.  At  A the 
angles  XAB  and  TAB  are  ob- 
served, and  at  B the  angles  AB  Y 
and  ABX.  Then  in  the  triangle 
XAB , 


BXA  = 180°  - XAB  - ABX, 


AX=AB^^. 

sin  BXA 


Also  in  the  triangle  TAB, 

BTA  = 180°  - YAB  - ABT, 


AY  — 


tsin  ABT 
AnBYA' 


Thus  AX  and  AT  are  known,  and  the  angle  included  be- 
tween them  is  XA  T — XAB  — YAB ; then  in  the  triangle 
XAT  the  angles  at  X and  Y can  be  found  by  either  of  the 
methods  of  Art.  1,  and  lastly  the  distance  XY.  As  a check  on 
the  work  the  sides  BX  and  BY  may  be  computed,  and  the  liue 
distance  ATFbe  again  found  from  the  triangle  A BY. 


ELEVATIOKS  AND  HEIGHTS. 


For  example,  let  it  be  required  to  find  the  horizontal  distance 
between  two  spires  X and  Y.  The  base  AB  is  laid  off  406.2 
feet  in  length,  and  the  measured  angles  are  XAB  = 83°  47', 
YAB  = 42°  32',  ABY  — 76°  52',  and  ABX  = 36°  20'.  Then 
the  side  BY  is  found  to  be  315.2  feet,  BX  to  be  466.83  feet, 
and  their  included  angle  is  40°  32'.  The  angles  BYX  and 
YXB  are  next  found  to  be  97°  26'  and  42°  02',  respectively 
Lastly,* the  required  distance  XY  is  306.0  feet. 

Prob.  8.  In  order  to  find  the  horizontal  distance  between  the 
tops  of  two  peaks  a base  line  5000  feet  long  was  laid  off.  At 
one  end  of  the  line  the  angles  between  the  base  and  the  peaks 
were  120°  and  50°,  at  the  other  end  of  the  line  they  were  95° 
and  40°.  Find  the  distance  between  the  peaks,  and  check  the 
computation. 

Art.  9.  Elevations  and  Heights. 

The  difference  in  level  between  two  points  on  the  ground 
which  are  accessible  is  usually  found  by  means  of  a leveling 
instrument  and  a graduated  rod.  The  level  is  placed  in  a 
horizontal  plane  by  means  of  its  bubble,  and  horizontal  sights 
are  taken  upon  the  rod  held  vertical  at  each  of  the  points. 
Thus  in  the  figure  to  find  the  difference  in  level  between  A and 


B the  level  is  placed  between  them;  the  rod  is  first  held  at  A, 
and  the  distance  a is  read  between  the  foot  of  the  rod  and  the 
point  where  the  horizontal  line  through  the  level  cuts  it,  the 
rod  is  next  moved  to  B and  the  distance  bx  is  there  read;  then 
the  difference  in  level  of  A and  B , or  the  elevation  of  A above 
By  is  bx  — a.  When  the  difference  of  level  between  two  points 
A and  G is  greater  than  the  length  of  the  rod,  the  level  is  set 
up  twice,  as  shown  in  Fig.  15;  then  the  difference  of  level  be- 
tween A and  G is  bx  — a -f-  c — b2.  This  process  may  be  con- 


28 


FUNDAMENTAL  PRINCIPLES. 


tinued  as  many  times  as  necessary,  and  tlie  difference  in  level 
between  the  initial  and  final  points  is  then  the  sum  of  the 
forward  readings  minus  the  sum  of  the  backward  readings. 


The  elevation  of  a point  is  its  height  above  sea  level  or  above 
some  datum  plane.  In  running  levels  it  is  customary  to  start, 
from  some  point,  called  a bench-mark,  whose  elevation  is 
known.  Thus,  in  Fig.  15,  let  the  point  A be  a bench-mark 
whose  elevation  is  328.72  feet,  and  let  the  reading  a he  0.93 
feet,  bi  he  10.84  feet,  b2  he  1.03  feet,  and  c be  11.47  feet.  Then 
the  elevation  of  B is  318.81  feet  and  the  elevation  of  C is 
308.37  feet. 


A^r 


The  height  of  an  inaccessible  point  is  usually  found  by  the 
help  of  vertical  angles  together  with  a measured  base  and 

certain  horizontal 
angles.  Let  it  be 
required  to  find* 
the  height  of  the 
top  of  the  flag- 
pole X above  the 
point  Y at  the 
base  of  the  build- 
ing. In  any  con- 
venient position 
let  a horizontal 
base  AB  be  meas- 
ured, also  let  the 
horizontal  angles 
CBA  and  BAG  be  measured  where  G is  a point  vertically 
below  X and  at  the  same  elevation  as  A ; in  reality  no  point 
G is  established,  but  these  angles  are  measured  by  pointing  the 
instrument  at  X,  the  angle  CBA  being  the  horizontal  projec- 
tion of  the  angle  XBA.  The  horizontal  angles  DBA  and 
BAD  are  likewise  measured  where  D is  a point  vertically 
above  Y.  At  A the  vertical  angles  XAG  and  YAD  are  also 
measured. 

In  the  triangle  ABC  two  angles  and  6ne  side  are  now  known, 
and  from  these  the  horizontal  line  A G is  computed.  Then 
in  the  right  triangle  A GX  the  side  AC  and  the  vertical  angle  at 


ERRORS  OF  MEASUREMENTS. 


29 


A are  known,  and  from  tliese  the  vertical  height  XC  is  com- 
puted. Again,  in  the  triangle  ABB  two  angles  and  one  side 
are  known,  from  which  the  horizontal  side  AD  is  found  ; then 
in  the  right  triangle  ADY the  vertical  side  BY  is  computed 
from  AB  and  the  vertical  angle  at  A.  Finally,  the  required 
height  XY  is  the  sum  of  XC  and  YB. 

As  an  example,  let  the  base  AB  be  314.62  feet,  CBA  = 40°  17', 
DBA  = 38°  22',  BAC  = 48°  40',  BAD  ±=  46°  57',  while  the  ver- 
tical angles  at  A are  XAC  — 37°  18'  and  YAB  = 5°  08'.  Then 
the  side  A C is 

AC=  314.62  = 203,46  feet, 

sin  91  03 

and  in  like  manner  AD  is  found  to  be  195.80  feet.  Then 

XC  — A. (7  tan  37°  18'  = 154.99  feet ; 

YD  — AD  tan  5°  08'  = 17.59  “ 

and,  lastly,  the  height  XY  is  154.99  + 17.59  = 172.6  feet,  the 
second  decimal  being  omitted,  as  it  is  probably  inaccurate. 

In  case  that  Y is  a point  on  the  building  above  the  level  of 
the  instrument  at  A,  as  may  often  happen,  then  XY  is  the 
d iff erence  of  XC  and  YD.  In  order  to  check  the  work  vertical 
angles  may  also  be  observed  at  B. 

Prob.  9.  In  order  to  find  the  difference  in  height  of  two 
peaks,  M and  X,  a base-line  AB  was  laid  off  5000  feet  long, 
and  the  horizontal  angles  BAM  = 120°  30',  BAX  = 49°  15', 
ABM  = 40°  35',  ABX  = 95°  07',  were  read.  At  A the  angle 
of  elevation  of  M was  17°  19',  and  the  angle  of  elevation  of  X 
was  18°  45'.  Compute  the  difference  in  height  of  tlie  two 
peaks. 

Art.  10.  Errors  of  Measurements. 

All  measurements  are  subject  to  errors  which  may  be  divided 
into  two  classes,  systematic  or  constant  errors,  and  accidental 
errors.  Systematic  errors  are  those  that  always  have  the  same 
value  under  the  same  circumstances,  being  due  to  known 
causes  ; for  example,  if  a 100-foot  chain  be  one  foot  too  long, 
all  measurements  made  with  it  will  be  one  per  cent  too  short. 
Accidental  errors  are  those  that  are  equally  likely  to  render  th§ 


30 


FUNDAMENTAL  PRINCIPLES. 


measurement  larger  or  smaller  than  the  true  value,  being  due 
to  the  combination  of  many  unknown  causes';  for  instance, 
variations  in  wind,  imperfection  of  eyesight,  and  other  similar 
causes  render  a measurement  too  great  or  too  small. 

Systematic  or  constant  errors  can  be  removed  from  measure- 
ments, when  their  causes  are  understood,  either  by  a proper 
method  of  observing  or  by  applying  proper  corrections  to  the 
numerical  results.  Methods  of  doing  this  for  both  linear  and 
angular  measures  will  be  given  in  the  following  chapters. 

After  all  the  systematic  errors  are  removed  the  numerical 
results  are  still  affected  by  the  accidental  errors.  As  these  are 
equally  likely  to  increase  or  decrease  the  true  value  of  the 
quantity  they  tend  to  balance  one  another,  and  hence  if  only 
one  measurement  be  made  it  must  be  accepted  as  the  most 
probable  value.  For  instance,  if  one  measurement  of  a line 
gives  618.5  feet,  after  the  systematic  errors  are  removed,  that 
value  must  be  taken  as  representing  the  true  value. 

When  several  measurements  of  a line  are  made  under  the 
same  conditions  each  has  the  same  degree  of  probability,  and 
hence  their  arithmetical  mean  is  to  be  taken  as  the  most  prob- 
able value  r for  example,  if  three  measures  of  a line,  made  in 
the  same  manner,  gives  618.5,  619.1,  and  618.9  feet,  there  is 
no  reason  for  preferring  one  to  the  other,  and  hence  one  third 
of  their  sum,  or  618.83  feet,  is  to  be  taken  as  the  most  probable 
length. 

If  the  three  angles  of  a triangle  are  measured  with  equal 
care  their  sum  should  be  180  degrees.  If  this  is  not  the  case 
the  results  are  to  be  adjusted  by  applying  one-third  of  the 
error  to  each  of  the  measured  angles.  So  with  a polygon  of 
n sides,  when  the  n interior  angles  are  measured,  their  sum 
should  equal  180n  — 360  degrees,  and  if  this  is  not  the  case 
one-nth  of  the  error  should  be  applied  to  each  of  the  measured 
values  in  order  that  their  sum  may  equal  the  theoretic  amount. 

When  the  sides  and  angles  of  a field  are  measured  the  sum 
of  the  northings  should  equal  the  sum  of  the  southings,  and 
also  the  sum  of  the  westings  should  equal  the  sum  of  the  east- 
ings. Owing  to  errors  in  measurement  these  conditions  will 


ERRORS  OF  MEASUREMENTS. 


31 


rarely  occur,  and  hence  an  adjustment  must  be  made,  as  ex- 
plained in  Art.  6,  to  remove  the  accidental  errors. 

When  three  angles  AOB,  BOG , AOG  are  measured  at  a 
station  0 with  equal  care,  the  sum  of  C 
AOB  and  BOG  should  equal  AOG.  If  \ B 

this  is  not  the  case  an  adjustment  must  \ y^ 

be  made  by  applying  one-tliird  of  the  \ .y 

error  to  each  angle.  For  example,  let  \ / 

the  measured  values  be  AOB  — 32°  16',  o A 

50(7=55°  43',  and  AOG  = ST  57';  Fis' 17- 

then  the  adjusted  values  are  AOB  — 32°  15'  20",  BOG  — 55° 
42'  20",  and  AOG  — 87°  57'  40  ',  which  exactly  satisfy  the  the- 
oretic condition.  It  is  always  advantageous  to  measure  the 
three  angles  even  if  only  two  are  required,  as  thus  a check  is 
furnished  on  the  work  and  opportunity  is  offered  to  eliminate 
the  accidental  errors  of  the  measurements. 

The  young  surveyor  should  always  bear  in  mind  that  the 
results  of  his  measurements  in  the  field  are  not  the  true  values 
of  the  quantities  which  they  represent,  but  only  approximate 
representations  of  the  true  values.  He  should  seek  to  secure 
the  greatest  degree  of  precision  consistent  with  the  tools  em- 
ployed and  the  end  in  view.  A large  part  of  the  land  surveys 
in  the  United  States  has  been  made  by  rough  and  imperfect 
methods,  but  the  time  has  now  come  when  precision  is  de- 
manded. Hence  care  must  be  taken  to  make  sufficient  meas- 
urements so  that  the  work  can  be  checked,  to  remove  all  sys- 
tematic sources  of  error,  and  finally  to  adjust  the  results  when 
possible  so  that  the  accidental  errors  maybe  largely  eliminated. 
In  precise  triangulation  work  the  adjustment  of  measurements 
is  especially  important,  and  the  principles  and  methods  for 
doing  this  constitute  a branch  of  science  known  as  the  method 
of  least  squares. 

Prob.  10.  At  a point  0 four  angles  are  measured  as  fol- 
lows : AOB  = 35°  07',  BOG  = 60°  43',  COB  = 22°  OP, 
AOB  — 117°  53'.  Find  their  adjusted  values. 


LAND  SURVEYING. 


3# 

CHAPTER  II. 

LAND  SURVEYING. 

Art.  11.  Chains  and  Tapes. 

The  chains  used  in  land  surveying  are  made  of  steel  wire 
and  have  the  joints  brazed  to  prevent  opening.  Iron  chains 
are  seldom  used,  being  heavier  and  in  every  way  inferior  to 
those  made  of  steel.  At  intervals  of  10  links  brass  tags  are 
fastened,  having  one,  two,  three,  or  four  points,  indicating 
distances  of  ten,  twenty,  thirty,  or  forty  links  from  either  end; 
the  middle  of  the  chain  is  marked  by  a round  tag.  The  chain 
is  provided,  at  either  end,  with  brass  handles  fastened  to  it  by 
a nut  and  screw  by  which  the  length  may  be  changed  a small 
amount.  The  length  of  the  chain  includes  the  handles.  In 
using  the  chain  care  must  be  taken  to  observe  whether  the  dis- 
tance is  greater  or  less  than  half  a chain,  as  forty  links  and 
sixty  links  are  marked  alike,  and  thirty  links  from  seventy 
links,  as  also  twenty  links  from  eighty  links,  must  be  carefully 
distinguished. 

The  chain  is  folded  by  bringing  the  49th  and  51st  links  to- 
gether, the  48th  and  52d  together,  and  so  on  until  the  ends  ar^ 
reached,  folding  links  equidistant  from  the  middle  together. 
To  unfold  the  chain,  hold  both  handles  in  the  left  hand  and 
with  the  right  hand  throw  it  horizontally  far  enough  so  that  it 
will  become  taut  before  it  falls. 

The  chain  possesses  some  advantages  over  the  tape  on  account 
of  its  weight  and  strength,  and  because  it  can  be  more  easily 
repaired.  In  chaining  through  brush  the  weight  of  the  chain 
is  serviceable  in  swinging  it  over  the  bushes  and  in  making  it 
straight  and  horizontal.  If  the  chain  is  broken,  a new  link 
may  be  put  in  by  the  surveyor. 

Steel  tapes  are  made  in  various  lengths  up  to  500  feet;  those 
having  lengths  of  50  feet  or  100  feet  are  generally  used  in 
land  surveying.  The  best  tapes  of  these  lengths  are  about  0 4 
inches  wide  and,  perhaps,  0.005  inches  thick;  they  are  gradu- 


CHAINS  AND  TAPES. 


33 


\ atecl  throughout  the  entire  length  into  hundredths  of  a foot,  and 
often  the  reverse  side  is  divided  into  rods  and  links.  These 
tapes  are  easily  broken,  anti  are  only  used  where  the  value  of 
the  land  warrants  very  careful  measurements;  they  rust  easily 
and  should  be  wiped  dry  after  using,  and  all  small  spots  of  rust 
removed  with  kerosene. 

Tapes  used  in  common  land  surveying  are  narrower  and 
thicker  than  those  described  above;  the  first  foot  from  either 
end  is  divided  into  tenths,  the  first  and  last  five  foot  spaces  are 
divided  into  feet,  and  the  tape  throughout  is  marked  every  five 
feet.  When  nickel-plated  these  tapes  require  much  less  at- 
tention to  keep  them  from  rusting  than  the  finer  grades.  In 
nearly  every  point  of  difference  between  such  a tape  and  the 
best  chain  the  comparison  is  in  favor  of  the  tape  ; one  great 
advantage  is  that  wear  does  not  increase  its  length  to  the  same 
degree  as  in  a chain. 

Metallic  tapes,  so  called,  are  made  of  cloth,  and  have  strands 
of  fine  brass  wire  interwoven  longitudinally.  They  are  divided 
throughout  into  tenths  of  a foot,  and  are  very  useful  in  making 
short  measurements  when  great  accuracy  is  not  required,  as  in 
finding  the  dimensions  of  buildings,  taking  offsets  to  locate 
paths,  brooks,  and  other  details  of  topography. 

To  use  the  tape  or  chain,  two  men  are  required,  called 
respectively  the  head  chainman  and  rear  chainman.  The  chain 
is  brought  into  the  line  and  made  level  with  the  rear  end  over 
the  first  point ; the  head  chainman,  by  means  of  a plumb-bob, 
finds  the  spot  directly  under  the  front  end  of  the  chain,  and 
marks  it  by  a nail  or  iron  pin  made  for  the  purpose.  This 
operation  is  repeated  till  the  end  of  the  line  is  reached. 

If  pins  are  used  there  should  be  eleven  of  them.  The  head 
chainman  places  a pin  at  the  front  end  of  the  chain,  and  this  is 
taken  up  by  the  rear  chainman  after  the  head  chainman  has 
placed  a second  pin.  When  the  last  pin  is  in  the  ground  the 
rear  chainman  delivers  his  ten  pins  to  the  head  chainman  and 
the  work  is  continued.  Each  delivery,  which  is  generally 
called  a tally,  thus  indicates  ten  chain  lengths. 

In  using  the  plumb-bob  with  the  chain,  it  is  best  to  stand 


34 


LAND  SURVEYING. 


facing  across  tlie  line  to  be  measured  ; the  string  is  Held,  against 
the  proper  point  on  the  chain  with  the  thumb  and  forefinger  of 
the  right  hand,  and  the  left  hand,  pressing  against  them,  helps 
in  stretching  the  chain.  The  head  chainman,  after  finding 
approximately  where  the  point  will  be,  should  carefully  clear 
away  all  leaves  and  grass,  and  prepare  a smooth  place  on  the 
ground,  so  that  a slight  touch  of  the  plumb-bob  may  be  suffi- 
cient to  mark  the  point. 

In  passing  along  the  line  the  rear  end  of  the  chain  is  allowed 
to  drag  along  the  ground,  and  just  before  it  reaches  the  pin 
the  head  chainman  is  notified  of  the  fact  by  some  preconcerted 
signal,  such  as  “ chain”  or  “ chain  out  ” ; much  time  can  be 
saved  by  stopping  the  head  chainman  at  just  the  proper  time. 

On  steep  slopes  it  is  best  to  chain  down  hill.  When  the 
difference  in  elevation  of  the  ground  along  the  line  is  more 
than  six  or  seven  feet  in  a hundred  feet,  the  head  chainman 
carries  his  end  of  the  chain  out  as  usual  and  puts  it  in  line;  he 
then  goes  back  to  a place  which  is  not  more  than  six  feet  lower 
than  the  rear  end  of  the  chain  and  proceeds  in  usual  manner, 
except  that  a part  instead  of  the  whole  of  the  chain  is  used. 
When  the  measurement  of  one  of  the  short  divisions  is  com- 
pleted, the  rear  chainman  holds  the  proper  division  over  the 
point  last  determined,  and  the  operation  is  repeated  till  the 
front  end  of  the  chain  is'reached.  It  is  unnecessary  to  record 
or  even  to  notice  the  lengths  of  the  divisions,  as  the  end  of  the 
chain  will  be  a chain’s  length  from  the  point  of  beginning. 
This  operation  is  called  “ breaking  the  chain.” 

Instead  of  using  the  plumb-bob,  the  horizontal  distance  is 
often  found  in  accurate  work  by  measuring  along  the  surface 
of  the  ground,  and  afterwards  determining  the  difference  in 
height  of  points  between  which  the  measurements  were  taken. 
The  length  along  the  chain  then  represents  the  hypothenuse  of 
a right  triangle,  of  which  required  distance  is  another  side. 

A chain  should  be  frequently  compared  with  a standard  laid 
off  on  a floor  or  pavement.  For  common  work  in  land  survey- 
ing, such  a standard  may  be  laid  off  by  a good  steel  tape  which 
has  not  been  used.  For  precise  work  in  cities  the  steel  tape 


THE  TRANSIT. 


35 


itself  should  be  standardized,  which  can  be  done  by  the  depart- 
ment of  Weights  and  Measures  of  the  U.  S.  Coast  and  (Geodetic 
Survey  at  Washington  (see  Art.  28). 

Many  surveyors  prefer  to  have  a chain  a little  longer  than 
the  standard  in  order  to  compensate  for  lack  of  level  and  for 
lateral  deviations.  In  good  work,  however,  these  sources  of 
error  should  be  avoided,  and  the  chain  should  agree  exactly 
with  the  standard.  If  a chain  is  too  long  the  measured  length 
of  a line  is  too  small;  thus,  if  the  length  824.5  feet  be  obtained 
by  a hundred-foot  chain  which  is  0.14  feet  too  long,  the  true 
length  of  the  line  is  8.245  (100  -f-  0. 14)  = 825.7  feet.  If  a chain 
is  too  short  the  measured  length  is  too  large  ; thus  if  the  length 
785.8  feet  be  obtained  by  a chain  which  is  0.07  feet  too  short, 
the  true  length  of  the  line  is  7.858  (100  — 0.07)  = 785.25  feet. 

Prob.  11.  A careless  surveyor  measured  a field  with  a 
hundred-foot  chain,  and  computed  the  area  to  be  8 acres,  12 
rods,  146  square  feet.  It  was  afterwards  found  that  the  chain 
had  lost  one  link,  so  that  its  true  length  was  only  99  feet.  If 
the  computations  of  the  surveyor  were  correct,  what  is  the  true 
area  of  the  field. 


Art.  12.  The  Transit. 

The  surveyor’s  transit  consists  primarily  of  two  parts ; the 
first,  called  the  alidade,  determines  the  line  of  sight,  and  the 
second,  called  the  limb,  affords  means  of  determining  the 
angular  deviation  of  this  line  from  any  other.  The  alidade,  in- 
cluding the  telescope,  the  magnetic  needle  with  its  graduated 
circle  and  the  vernier,  is  attached  to  a vertical  spindle,  and 
may  be  revolved  while  the  limb  remains  stationary.  The  hori- 
zontal circle  composing  the  limb  is  graduated  into  degrees,  and 
sometimes  into  thirty  minute  or  twenty  minute  spaces,  and 
numbered  from  zero  to  360  degrees  in  both  directions.  The 
limb  is  mounted  upon  a hollow  cylindrical  annulus  which  sur- 
rounds the  spindle  of  the  alidade.  The  instrument  is  supported 
by  three  legs,  called  the  tripod,  which  are  fastened  together  at 
the  top  by  the  tripod  head. 

The  device  used  to  measure  fractional  amounts  of  the  divi- 
sions of  the  limb  is  called  a vernier.  Verniers  are  used  either 


36 


LAND  SURVEYING. 


on  straight  or  circular  scales,  the  former  being  employed  on 
level  rods  and  the  latter  on  transits.  In  Fig.  18  is  shown  a 
vernier  for  a straight  scale,  where  the  length  of  the  vernier  is 
the  same  as  the  length  of  nine  spaces  of  the  limb.  The  vernier 
itself  is  divided  into  ten  equal  parts.  Let  a be  the  length  of 


r 

Scale  or  Limb 

1,1  ill  1 1 

f 

rCT  i r iii  i i t 

1 I Vernier 

Scale  or  i Limb 

-i'-Ft'-t'-i 

1 Vernier 


Fig.  18. 

one  space  on  the  limb,  and  b the  length  of  one  space  on  the 
vernier.  On  a level  rod  a is  T^<jth  of  a foot,  then  b is  j^th  of 

^th  of  a foot,  hence 

a —b  — — yo9o  o — iirbo  I 

and  thus  the  space  between  the  first  division  of  the  limb  and 
the  first  division  of  the  vernier  in  Fig.  18  is  of  a foot,  or 
one-tenth  of  a space  of  the  limb. 

If  the  vernier  in  the  first  diagram  of  Fig.  18  is  moved  until 
its  first  division  coincides  with  the  first  division  of  the  limb  a 
distance  of  a or  ToVo  has  been  passed  over.  If  the  third 
divisions  coincide,  as  the  second  diagram,  the  vernier  lias 
moved  a distance  of  -f^a  or  Tq30D  feet.  Thus  in  moving  the  ver- 
nier fractional  parts  of  the  smallest  space  of  the  limb  are  read 
with  precision  by  noting  what  division  of  the  vernier  coincides 
with  a division  of  the  limb. 

If  the  length  of  the  vernier  is  equal  to  19  spaces  of  the  limb 
and  it  is  divided  into  20  parts,  the  distance  a — b will  be  one- 
twentieth  of  one  space  of  the  limb,  or  a degree  of  precision 
twice  as  high  as  before.  Hence  a general  rule  for  finding  the 
smallest  amount  indicated  by  the  vernier  is  this  : Divide  the 
value  of  the  smallest  space  of  the  limb  by  the  number  of  spaces 
on  the  vernier. 

A vernier  can  be  also  made  by  making  its  length  equal  to  11 


THE  TRANSIT. 


37 


spaces  of  tlie  limb  and  dividing  it  into  10  equal  parts,  or  by- 
making  its  length  equal  to  21  spaces  of  the  limb  and  dividing 
it  into  20  parts.  Such  an  arrangement  is  called  a retrograde 
vernier,  and  is  not  commonly  used. 

The  verniers  used  on  transits  are,  of  course,  circular  instead 
of  straight,  and  the  divisions  on  the  limb  are  degrees  and  frac- 
tions of  degrees  instead  of  feet,  but  the  principles  do  not  differ 
from  those  stated  above.  Such  verniers  are  usually  made 
double  for  convenience  in  reading  angles  in  either  direction. 
Such  a vernier  is  shown  in  Fig.  19.  Here  it  is  seen  that  the 
zero  point  on  the  vernier,  in  moving  from  the  right  to  the  left, 
has  passed  the  point  a , which  is  66°  30',  and  is  at  b.  By  using 


the  vernier  it  is  possible  to  measure  the  space  a b.  In  the 
figure  the  limb  is  divided  into  thirty  minute  spaces,  the  ver- 
nier is  of  the  same  length  as  twenty-nine  of  these  spaces,  and 
is  divided  into  thirty  spaces.  Hence  the  smallest  amount  in- 
dicated by  such  a vernier  will  be  the  difference  between  the 
lengths  of  a space  on  limb  and  on  the  vernier,  or  one  minute. 
By  referring  to. the  figure  it  is  seen  that  the  fourth  division  on 
the  vernier  to  the  left  of  zero  coincides  with  one  on  the  limb, 
hence  the  zero  point  has  moved  four  minutes  after  passing  the 
point  cty  and  the  reading  is  66°  30'  -f  04'  or  66°  34'. 

In  using  the  double  vernier  the  beginner  may  be  in  some 
doubt  as  to  which  part  to  use.  This  can  be  guarded  against 
by  reading  that  side  which  is  farthest  away  from  zero  on  the 
limb,  in  the  direction  that  the  vernier  has  been  turned. 

The  precision  of  the  work  done  by  an  instrument  depends  as 
much  upon  the  care  taken  of  it  as  upon  its  original  excellence. 


38 


LAND  SURVEYING. 


In  carrying  tlie  transit  to  and  from  work,  care  must  be  taken 
that  the  tripod  is  firmly  attached  ; the  telescope  should  be 
turned  in  line  with  the  axis  of  the  instrument,  but  not  too 
rigidly  clamped  ; the  cap  should  be  placed  over  the  objective 
and  the  needle  lifted  from  the  centre  pin.  Tlie  instrument, 
while  being  carried,  is  held  on  the  shoulder  by  the  hand  just 
in  front  with  the  elbow  close  to  the  side  ; in  this  way  there  is 
more  freedom  of  movement  and  the  least  liability  to  accident. 

In  setting  up  the  instrument  it  is,  in  most  cases,  better  to  put 
two  legs  down  hill  and  one  leg  up  hill.  The  instrument  is 
lifted  bodily  and  set,  as  nearly  as  may  be,  over  the  point,  with 
the  plates  parallel  and  horizontal.  In  bringing  the  transit  into 
exactly  the  required  position  it  is  only  necessary  to  remember 
that  the  plumb-bob  will  follow  the  direction  in  which  either 
leg  is  made  to  move — toward  it  or  away  from  it  according  as 
the  leg  is  carried  out  or  in.  It  is  not  well  to  force  the  tripcd 
feet  further  into  the  ground  than  is  necessary  for  rigidity  ; 
some  tripods  are  wisely  furnished  with  lugs  to  receive  tLe 
pressure  from  the  foot ; thus  the  tripod  head  is  relieved  of 
much  unnecessary  strain. 

After  the  instrument  has  been  set  up  with  the  plumb-bob 
over  the  point,  the  next  step  is  to  level  the  plates.  The  in- 
strument is  first  turned  so  that  the  bubble  tubes  are  parallel 
to  the  lines  through  the  two  opposite  leveling  screws  ; it  is 
then  leveled  by  turning  the  screws  in  opposite  directions;  this 
will  be  accomplished  when  the  thumbs,  in  turning,  move 
either  toward  or  from  each  other.  The  bubble  will  be  seen  to 
move  in  the  direction  in  which  the  left  thumb  moves.  After 
all  the  leveling  screws  are  brought  to  a bearing  on  the  plates 
by  turning  one  screw  in  each  pair,  they  should  only  be  turned 
in  pairs  and  in  opposite  directions;  in  this  way  the  bearing 
upon  the  plates  will  be  preserved  and  the  screws  and  plates 
will  not  become  strained. 

Suppose  the  transit  to  be  set  over  the  point  0 in  Fig.  17  and 
that  it  is  desired  to  measure  the  horizontal  angle  AOB.  Tlie 
telescope  is  directed,  with  tlie  vernier  clamped,  toward  either 
of  the  points  B or  A,  and  the  limb  clamped  ; the  vernier  is 


THE  TRANSIT. 


39 


then  read  and  undamped,  and  the  telescope  is  directed  toward 
,•  the  other  point,  the  alidade  clamped,  and  the  vernier  read 
again.  It  is  evident  that,  as  the  vertical  plane  of  the  telescope 
and  the  vernier  are  relatively  immovable,  the  angular  distance 
passed  over  by  the  zero  point  on  the  vernier  and  by  the  plane 
of  the  telescope  are  the  same,  or  the  angle  AOB.  Hence,  to 
measure  an  angle,  readings  of  the  vernier  are  made  before  and 
after  the  angle  is  turned,  and  the  difference  is  taken.  In  or- 
dinary work  it  is  usual  to  set  the  vernier  at  zero  before  turning 
the  angle,  in  which  case  the  reading  after  the  second  sight  has 
been  taken  is  the  angle  itself. 

It  is  only  necessary  to  follow  the  above  directions  to  cor- 
rectly measure  any  angle,  but  the  operation  can  seldom  be  done 
by  a beginner  so  that  no  errors  are  involved.  It  is  readily  seen 
that  the  accuracy  of  the  measurement  of  an  angle  depends 
upon  the  following  : 

The  adjustment  of  the  transit. 

Setting  the  instrument  over  the  exact  point  it  is  desired  to 
have  it  occupy. 

The  reading  of  the  vernier. 

The  bisection  of  the  points  toward  which  the  telescope  is 
directed. 

The  movement  of  the  alidade  due  to  defects  in  clamping. 

In  land  surveying  where  angles  are  only  read  to  the  nearest 
minute  these  errors  should  be  made  as  small  as  possible  by 
seeing  that  the  transit  is  in  adjustment,  that  it  is  set  over  the 
exact  centre  of  the  station,  that  the  vernier  is  accurately  read, 
that  the  signals  sighted  upon  are  correctly  placed  and  truly 
bisected,  and  that  care  is  taken  in  using  the  clamps.  Direc- 
tions for  adjusting  a transit  are  given  in  Art.  27,  but  a beginner 
should  never  attempt  to  make  them  until  he  has  used  the  instru- 
*ment  sufficiently  to  become  thoroughly  acquainted  with  ali 
the  manipulations. 

In  precise  work  where  angles  are  needed  to  fractions  of  a 
minute  the  last  three  sources  of  error  mentioned  above,  as  well 
as  some  others,  may  be  largely  eliminated  by  the  method  of 
repetitions  described  in  Art.  28.  In  land  surveying  repetitions 
are  unnecessary,  but  it  will  be  well  to  check  each  angle  by 


40 


LAND  SURVEYING, 


measuring  also  its  explement.  Thus,  if  the  angle  AOB  is  read 
by  pointing  first  on  A and  then  on  Bf  let  the  angle  BOA  be  read 
by  pointing  first  on  B and  then  on  A ; the  sum.  of  the  two 
angles  should  be  300°  00'. 

An  engineer’s  transit  mainly  differs  from  a surveyor’s  transit 
in  having  a vertical  arc  and  a level  bubble  attached  to  the  tele- 
scope for  the  determination  of  heights  and  elevations.  Some 
engineers’  transits  have  verniers  reading  to  half-minutes,  while 
transits  for  triangulation  work  sometimes  read  to  twenty 
seconds  or  to  ten  seconds0 

Prob.  12.  If  the  limb  is  divided  into  20-minute  spaces,  show 
how  the  vernier  must  be  made  in  order  to  read  one  minute? 
in  order  to  read  20  seconds?  Give  diagrams  of  these  verniers. 

Art.  13.  The  Magnetic  Needle. 

Most  of  the  early  land  surveys  of  the  United  States  were 
made  by  the  compass.  The  compass  is  an  instrument  like  the 
surveyor’s  transit,  but  without  graduated  limb  and  telescope  ; 
the  place  of  the  latter  is  supplied  by  vertical  sights,  while 
angles  are  read  by  bearings  of  the  magnetic  needle.  All  the 
remarks  here  made  regarding  the  magnetic  needle  apply 
equally  to  the  compass  and  to  the  transit,  although  in  the  case 
of  the  transit  the  needle  is  used  less  than  the  graduated  limb 
and  vernier. 

The  compass  plate  is  usually  graduated  to  half-degrees  ; the 
north  and  south  points,  lettered  N and  S}  -are  marked  0°,  and 
the  graduation  runs  from  each  in  both  directions  to  the  east 
and  west  points  which  are  marked  90°.  The  letters  E and  W 
are,  however,  on  the  west  and  east  sides  respectively,  of  the 
compass  plate,  in  order  that  the  direction  of  a line  as  read  from 
the  end  of  the  needle  may  agree  with  its  actual  direction.  The 
direction  of  a line  as  determined  by  the  needle  is  called  its 
magnetic  bearing.  The  bearing  is  expressed  by  two  of  the 
letters  _ZV,  E , S,  or  W,  with  the  number  of  degrees  which  the 
line  varies  from  the  magnetic  meridian  ; thus  JV  35°  E , which 
is  read  north  thirty-five  degrees  east,  means  a line  whose  direc  - 
tion is  thirty-five  degrees  east  of  north  ; also  8 70°  W indicates 


THE  MAGNETIC  NEEDLE. 


41 


a line  whose  magnetic  direction  is  seventy  degrees  west  of 
south. 

When  the  bearings  of  several  lines  are  taken  at  the  same 
point  the  angles  between  them  are  known.  For  example,  let 
the  bearing  of  AC  be  J8i°  E , and 
that  of  AD  be  N 46°  E , then  the  angle 
BAD  is  37  j degrees.  Also  if  the  bear- 
ing of  AF  be  8 52^°  E,  then  the  angle 
DAF  is  81£  degrees.  The  student 
should  deduce  his  own  rule  for  find- 
ing the  angle;  from  the  bearings  by 
drawing  figures  for  a few  special 
cases.. 

When  the  bearings  of  several 
courses  are  given  the  angles  between  them  are  also  known. 
Thus,  in  Fig.  21  let  the  bearing  of  AB  be  N 42°  E , and  that 
of  BCbe  8 29£°  E ; then  the  angle  ABC  is  71i°.  Here  it  is 
best  to  reverse  the  bearing  of  the  first  line,  and  thus  consider 
both  as  taken  at  the  point  B where 
the  bearing  of  BA  is  8 42°  W. 

The  magnetic  needle  is,  at  the  best, 
a rough  and  imperfect  tool  for  meas- 
uring angles  or  for  determining  the 
directions  of  lines.  The  bearings  can 
be  read  to  quarters  or  eighths  of  a 
degree,  but  owing  to  the  variations  to 
which  the  needle  is  subject,  a line  will 
have  different  bearings  at  different 
times.  The  magnetic  meridian  at  most  places  deviates  from  the 
true  meridian,  and  the ’angle  between  them  is  called  the  declina 
tion  of  the  needle.  On  the  Atlantic  coast  of  the  United  States 
the  declinatio  i is  to  the  west  of  the  true  meridian,  while  on  the 
Pacific  coast  it  is  to  the  east,  but  its  amount  is  very  different  in 
different  places,  as  will  be  seen  from  the  isogonic  map  of  the 
United  States  for  1900  inserted  at  page  128  of  this  Handbook. 
An  isogonic  line  is  a curve  passing  through  all  places  which 
have  the  same  magnetic  meridian.  Thus  in  1900  the  line  of 
zero  declination  passes  near  Columbus,  Ohio,  and  Charleston, 


42 


LAND  SURVEYING. 


S.  C.,  and  during  that  year  the  magnetic  meridian  coincides 
with  the  true  meridian  at  all  places  on  that  line.  These 
isogonic  lines  are  now  slowly  shifting  westward. 

The  secular  variation  of  the  magnetic  needle  is  an  oscillatory 
movement  by  which  the  declination  varies  hack  and  forth  from 
a mean  value.  The  time  of  this  oscillation  in  the  United  States 
is  between  two  and  three  centuries,  hut  a complete  cycle  has 
not  yet  been  observed.  For  example,  at  New  York,  N.  Y.,  the 
early  observations  indicate  that  in  1657  the  needle  was  at  its 
extreme  western  declination  of  9^  degrees  ; this  slowly  de- 
creased so  that  about  1795  it  reached  the  minimum  value  of  44 


degrees ; during  the  nineteenth  century  it  has  slowly  increased 
and  will  probably  reach  the  extreme  western  declination  about 
1933,  the  total  period  of  the  cycle  thus  being  276  years.  Fig.  22 
shows  clearly  to  the  eye  these  variations  in  declination,  as  also 
those  at  Washington,  D.  C. , where  the  minimum  value  was  ob- 
served in  1810,  while  the  maximum  will  probably  occur  in  1927. 

The  value  of  the  declination  for  1900  may  be  ascertained 
approximately  from  the  isogonic  map  above  referred  to.  Its 
value  at  any  date  may  be  found  for  a large  number  of  places 
by  means  of  the  formulae  deduced  by  the  U.  S.  Coast  and 
Geodetic  Survey,  and  given  in  the  report  for  1895,  pages  167 
to  320.  For  example,  the  formula  for  Bethlehem,  Pa.,  is 

D ==  5°. 27  + 3°. 05  sin  (1°.46  - 34°. 8), 


THE  MAGNETIC  NEEDLE. 


43 


in  which  D denotes  west  declination  and  m is  the  number  of 
years  counted  from  Jan.  1,  1850.  If  it  be  required  to  find  tbe 
declination  for  April  30,  1887,  tbe  value  of  m is  37.8  years,  and 
then, 

D = 5°. 27  4-  3°. 05  sin  19°. 7 ==  6°. 50  west. 

From  tbe  formula  also  can  be  found  tbe  values  and  tbe  dates 
of  tbe  maximum  and  minimum  declinations.  Tbe  greatest 
declination  will  occur  when  tbe  angle  l°.46m  — 34°. 8 equals  90c, 
as  tbe  sine  is  then  unity  ; this  gives  D = 8°. 32  and  m — 85  5 
years,  so  that  tbe  time  of  this  occurrence  will  probably  be  in 
the  year  1935.  The  least  declination  obtains  when  tbe  sine  is 
minus  unity,  and  this  gives  D — 2°. 22,  and  m — — 37.8,  which 
corresponds  to  the  year  1812. 

The  daily  variation  of  tbe  needles  is  a small  oscillation  rang- 
ing from  5 to  10  minutes  in  different  seasons  and  places.  It  is 
smaller  in  the  winter  than  in  the  summer,  and  less  in  tbe 
southern  part  of  the  United  States  than  in  tbe  northern  part. 
Soon  after  sunrise  tbe  north  end  of  tbe  needle  is  at  its  most 
easterly  deviation  from  tbe  magnetic  meridian.  A westerly 
motion  then  begins,  and  about  half-past  ten  o’clock  it  coincides 
with  that  meridian  ; tbe  westerly  motion  continues  until  about 
half-past  one  o’clock  in  tbe  afternoon  when  tbe  most  westerly 
deviation  is  reached.  Tbe  easterly  motion  is  then  slowly 
resumed  and  by  the  next  morning  tbe  needle  again  reaches  its 
most  easterly  deviation.  Table  III,  at  tbe  end  of  this  book, 
gives  the  mean  values  of  tbe  daily  variation  for  each  hour  of 
tbe  day  and  each  month  of  tbe  year  at  Philadelphia,  Pa.,  as 
also  instructions  for  finding  it  for  other  places  in  tbe  United 
States. 

In  addition  to  tbe  secular  and  daily  variations  tbe  magnetic 
needle  is  also  subject  to  an  annual  variation  of  about  1£  min- 
utes, and  to  other  smaller  variations  caused  by  tbe  moon  and 
sun.  Magnetic  storms  cause  sudden  variations  of  considerable 
amount.  These  minor  variations,  however,  are  of  little  im- 
portance in  land  surveying,  compared  to  tbe  local  attraction 
that  is  liable  to  occur  in  rocky  regions  and  which  often  causes 
discrepancies  of  several  degrees  in  tbe  bearings  of  a line  taken 
at  points  only  a few  hundred  feet  apart.  Tbe  method  of 


44 


LAND  SURVEYING. 


eliminating  tlie  effect  of  local  attraction  is  explained  in  tlie 
next  article. 

Prob.  13.  The  formula  for  tlie  west  declination  at  New 
Brunswick,  N.  J.,  is 

D = 5°. 11  + 2°. 94  sin  (1°.30 m + 4°.2). 

Find  tbe  values  of  the  maximum  and  minimum  declinations 
with  the  dates  of  their  occurrence.  Find  also  the  probable 
value  of  the  declination  on  June  15,  1896. 

Art.  14.  Field  Work. 

The  field  work  in  land  surveying  may  be  divided  into  two 
classes,  original  surveys,  and  resurveys.  The  first  class  in- 
cludes not  only  the  case  of  lands  opened  for  the  first  time  for 
settlement,  but  also  the  staking  out  and  division  of  lands,  and 
all  surveys  which  are  made  without  particular  reference  to  for- 
mer records.  Resurveys,  on  the  other  hand,  are  those  made  to 
trace  boundaries  that  have  been  lost,  and  they  require  the 
knowledge  of  the  former  work  which  are  either  stated  in  deeds 
on  maps,  or  in  the  records  of  towns  or  counties.  In  both 
cases  the  field  work  requires  the  measurement  of  such  lines 
and  angles  as  will  enable  a complete  map  of  the  property  to 
be  made,  and  the  areas  of  the  several  portions  to  be  computed. 

A field  party  usually  consists  of  three  or  four  men,  the  sur- 
veyor who  reads  the  angles  or  bearings  and  takes  the  notes, 
two  chainmen,  and  perhaps  an  axman  who  sets  the  necessary 
stakes  and  poles  and  also  assists  with  the  tape.  The  poles 
which  are  used  for  ranging  out  the  lines  and  to  sight  upon  in 
measuring  angles  are  generally  about  an  inch  in  diameter, 
about  eight  feet  long,  each  alternate  foot  being  painted  red 
and  white,  and  they  are  pointed  with  steel  to  enable  them  to 
be  easily  set  in  the  ground.  In  surveying  a field  it  is  an  old 
custom  for  the  party  to  go  around  the  boundaries  “ in  the  di- 
rection of  the  sun,”  that  is,  so  as  to  keep  the  field  on  the  right 
hand.  The  bearings  of  lines  can  thus  be  written  on  a sketch 
in  a natural  order  around  the  entire  circuit. 

It  frequently  happens  that  a surveyor  is  obliged  to  employ  ns 
chainmen  men  who  have  had  no  experience  in  such  work.  In 


FIELD  WORK. 


45 


this  event  it  is  well,  even  after  having  given  them  full  instruc- 
tions, that  he  should  be  constantly  with  them  for  several  hours 
in  order  to  ensure  that  the  proper  degree  of  precision  shall  be 
attained.  Chaining  indeed  is  far  more  difficult  to  do  accu- 
rately than  is  the  measurement  of  angles. 

The  point  where  a transit  is  set  for  the  purpose  of  reading 
angles  is  called  a station.  In  the  survey  of  a field  the  corners 
are  also  often  called  stations,  these  being  the  initial  points 
from  which  the  linear  measurements  are  taken.  A line  whose 
bearing  is  known  is  frequently  called  a course. 

If  the  surveyor  is  provided  with  a transit  it  is  advised  that 
angles  should  be  always  measured,  and  only  such  bearings  be 
taken  as  are  necessary  to  check  the  work  or  to  verify  former 
records.  If  he  has  only  a compass  the  bearings  of  the  lines  must 
be  taken,  but  care  should  be  exercised  to  avoid  the  errors  due 
to  local  attraction.  Fortunately  the  influence  of  this  can  be 
eliminated  by  always  reading  the  back  bearings  of  lines  as 
well  as  their  forward  bearings.  In  doing  this  the  instrument 
should  be  set  at  the  ends  of  the  lines  so  that  the  back  bearing 
of  one  line  and  the  forward  bearing  of  the  next  one  may  be 
read  at  the  same  station.  The  bearings  at  one  point  being  as- 
sumed to  be  correct,  all  the  others  can  then  be  adjusted  so  as  to 
be  relatively  correct. 

As  an  example  of  the  elimination  of  the  effect  of  local  attrac- 
tion let  the  bearing  of  AB  be  taken  at  A in  Fig.  9,  and  also 
the  back  bearing  of  EA\  then  at  B let  the  bearings  of  BA  and 
BG  be  taken,  and  so  on.  Let  the  results  obtained  be  those 
which  are  given  in  the  second  and  third  columns  of  the  table. 


Course. 

Bearing. 

Back  Bearing. 

Adjusted  Bearing. 

Azimuth. 

AB 

N 37°  15'  E 

S 38°  00'  W 

N 37°  15'  E 

37°  15' 

BC 

S 78  08  E 

N 77  45  W 

S 78  53  E 

101  07 

CD 

S 33  45  W 

N 33  15  E 

S 32  37  W 

212  37 

DE 

N 14  37  W 

S 15  30  E 

N 15  15  W 

314  45 

EA 

N 82  30  W 

S 82  15  E 

N 82  15  W 

277  45 

Now  assume  that  there  is  no  local  attraction  at  A , then  the 
bearing  of  AB  and  the  back  bearing  of  EA  are  correct.  To 
adjust  the  other  values  proceed  in  order  from  A to  B\  at  B the 


46 


LAND  SURVEYING. 


result  38°  00'  is  45'  too  large,  lienee  45'  must  be  subtracted 
•from  all  SW  and  NE  lines  starting  from  B and  the  same 
amount  must  be  added  to  all  SE  and  NW  lines;  thus  the  ad- 
justed bearing  of  BG  is  78°  53'.  Next  the  result  77°  45'  taken 
at  G is  seen  to  be  1°  08'  too  small,  and  this  must  be  applied  to 
the  forward  bearing  of  GD,  giving  the  adjusted  bearing  as 
S 32°  37'  W.  Thus  proceeding,  the  adjusted  bearing  of  EA 
comes  out  N 82°  15'  W,  and  this,  being  the  reverse  of  the  back 
bearing  taken  at  A,  is  a check  on  the  correctness  of  both  the 
field  work  and  the  adjustment. 

The  azipiuth  of  each  line  is  easily  found  from  its  adjusted 
bearing.  If  the  meridian  be  taken  to  correspond  with  the 
magnetic  meridian  the  results  given  in  the  last  column  of  the 
table  are  the  azimuths.  They  are  found  by  adding  or  sub- 
tracting each  bearing  either  to  or  from  180°  or  360°,  as  the  case 
may  require. 

The  interior  angles  of  a field  are  readily  computed  either 
from  the  adjusted  bearings  or  from  the  azimuths  of  the  lines. 
It  is,  however,  no  proof  of  the  correctness  of  the  field  work  if 
the  sum  of  these  angles  equals  the  proper  theoretic  sum,  for 
it  will  be  found  that  any  bearings  whether  correct  or  incor- 
rect will  give  the  correct  amount.  On  the  other  hand  if  the 
angles  be  measured  in  the  field  with  the  transit,  a valuable 
check  is  obtained^  by  taking  their  sum  which  will  only  equal 
the  theoretic  sum  in  very  good  work.  In  such  cases  if  no 
serious  error  is  thought  to  exist  the  observed  values  should  be 
adjusted  by  the  method  of  Art.  10. 

One  of  the  most  important  details  of  the  field  work  is  the 
keeping  of  the  notes.  Nearly  every  surveyor  has  a system  of 
his  own  for  recording  the  measurements  taken  in  the  field,  so 
no  one  method  can  be  said  to  be  the  standard  ; the  essential 
point  is  that  they  shall  be  readily  legible  to  any  person  who  is 
to  use  them.  Better  results  will  probably  be  obtained  by  mak- 
ing a sketch  in  the  field  book,  showing  objects  in  their  relative 
positions  and  having  the  dimensions  to  be  used  in  plotting 
marked  on  the  sketch  itself,  than  by  a more  elaborate  system 
of  symbols  and  abbreviations. 

If  the  survey  covers  but  a small  area,  as  one  or  two  lots  of 


SURVEY  OF  A FARM. 


47 


town  property,  all  tlie  notes  should  be  recorded  on  one  sketch, 
which  may,  to  make  the  scale  larger,  be  extended  across  two 
pages.  In  the  survey  of  a large  tract  it  will  be  better  to  devote 
a page  to  one  course;  repeating,  as  the  leaves  are  turned,  part 
of  the  notes  of  one  page  on  the  next. 

The  notes  should  be  made  with  a medium  hard  pencil  and  a 
straight-edge  be  used  in  drawing  all  lines  intended  to  be  straight. 
All  writing  should  be  in  upright  capitals,  and  no  script  should 
be  used.  Distances  along  the  line  are  usually  inclosed  in  a 
circle  or  parenthesis,  and  are  written  on  a line  perpendicular 
to  the  base.  It  will  be  generally  more  convenient  to  begin  the 
notes  at  the  foot  of  the  page,  as  by  so  doing  one  can  glance 
from  the  book  to  the  field  and  see  corresponding  lines  having 
the  same  direction  and  in  front.  Samples  of  field  notes  are 
given  in  Art.  15.  The  best  books  for  notes  have  both  sides  of 
the  leaves  ruled  alike  with  light-blue  lines  into  squares  about 
an  eighth  of  an  inch  on  a side.  Such  books  are  substantially 
bound  in  leather  and  cost  about  fifty  cents. 


Prob.  14.  Find  the  adjusted  bearings  of  the  sides  of  the 
following  field,  assuming  the  bearing  of  BG  to  be  correct. 


Course. 

Bearing. 

AB 

S 12°  15'  W 

BG 

N 76  45  W 

GD 

N 12  15  W 

BE 

N 47  37  W 

EF 

N 24  30  E 

FA 

S 75  15  E 

Also  compute  the  area  of  the 


Back  Bearing. 

Length 
in  Chains. 

N 12°  30'  E 

5.62 

S 76  45  E 

3.28 

S 12  07  E 

2.24 

S 48  00  E 

3.05 

S 24  15  W 

2.29 

N 75  00  W 

6.40 

in  acres,  roods,  and  rods. 


Art.  15.  Survey  of  a Farm. 

' In  Fig.  23  is  given  a sketch  of  the  farm,  a survey  of  which 
is  required.  The  farm  is  seen  to  comprise  three  divisions 
separated  from  each  other  by  fences,  and  it  is  desired  to  locate 
the  interior  division  lines  as  well  as  the  boundaries,  and  also 
to  mark  the  edge  of  the  wood-land  and  the  course  of  the 
brook. 

The  principal  lines  of  the  survey,  usually  called  traverse- 


48 


LAND  SURVEYING, 


1. 


LINE  TREE  / 


Land  of  john 

--'  S* 


Land  of  E.  D.  Parker 


Survey  of  Geo.  e.  William’s  Farm. 

Riverside,  Pa.  MajT,  1894. 

A,  C.  Thomas,  Surveyor 
^ F,  H>  Carter  & M.  T.  Miller, 
Assistants, 


Fig.  23. 


STAKE  & STONES 


SURVEY  OF  A FARM. 


49 


lines,  are  measured  outside  or  inside  tlie  boundaries  according 
to  the  degree  of  difficulty  presented  by  the  two  sides;  thus  it 
is  natural  that  the  measurement  of  the  line  AB  should  be 
easier  along  the  highway  than  along  the  inside  of  the  fence, 
also  EF  is  more  easily  measured  outside  the  boundary,  as  the 
ground  there  is  clear  of  trees. 

Besides  the  outside  polygon,  ABGDEFGHIJK , the  sec- 
ondary traverse  GNOPQG  is  run  to  locate  the  edge  of  the 
woods,  and  WVPBSTU  to  define  the  course  of  the  brook  ; 
perpendicular  offsets  to  the  boundary  lines  are  measured 
wherever  these  make  a decided  change  in  direction. 

The  manner  of  keeping  the  field-notes  is  shown  in  the 
following  sketches.  On  the  first  page  of  the  note-book  is  a 
general  outline  of  the  work  similar  to  that  in  Fig.  23.  The 
location  of  the  farm,  the  names  of  the  owners  of  this  and  of 
abutting  property,  should  be  ascertained  and  recorded,  and  also 
the  character  of  the  boundaries,  whether  wooden  fence,  stone- 
wall, hedge,  or  imaginary  line.  The  names  of  the  surveyor 
and  all  his  assistants,  and  the  date  upon  which  the  work  was 
done,  should  never  be  omitted.  On  the  second  and  succeeding 
pages  of  the  note-book  are  the  notes  of  the  traverses  (Figs. 
24-31).  These  are  made  by  beginning  at  the  bottom  of  the 
page  and  working  upward,  so  that  the  surveyor  always  has  the 
objects  to  be  recorded  in  the  same  relative  position  as  the 
sketches. 

The  survey  is  begun  by  setting  the  instrument  over  A and 
selecting  stations  B and  L.  The  interior  angle  BAL  is  read 
and  recorded  on  the  margin  of  the  page  opposite  A,  and  as  a 
check  on  this  reading  the  exterior  angle  is  also  measured  and 
written  under  the  first.  If  the  sum  of  the  two  angles  is  within 
one  minute  of  360  degrees  the  first  angle  is  recorded  on  an  arc 
.between  BA  and  AL , as  shown  in  Fig.  24;  if  such  agreement 
does  not  occur  the  angles  should  be  observed  again.  The 
fence-corner  opposite  A is  located  by  the  angle  between  it  and 
AB  from  A,  and  by  the  distance  from  A.  The  width  of  the 
highway  is  measured  and  the  station  is  referenced  as  explained 
below.  The  house  is  located  by  measuring  along  the  line  AB 
to  the  points  where  perpendiculars  to  that  line  would  pass 


50 


LAND  SURVEYING, 


SURVEY  OF  A FARM, 


51 


4. 


5. 


Fig.  26. 


Fig.  27. 


52 


LAND  SURVEYING. 


6. 


9. 


Fence 


SURVEY  OF  A FARM. 


53 


7.  8. 


54 


LAND  SURVEYING. 


through  two  corners  of  the  building;  the  lengths  of  the  per- 
pendiculars are  ascertained  and  the  dimensions  of  the  house 
measured.  The  distances  from  A along  the  traverse  to  the 
perpendiculars  are  recorded  in  circles  just  opposite  them,  and 
the  lengths  of  the  offsets  are  written  along  those  lines.  The 
notes  of  the  location  of  the  barn  are  taken  and  recorded  in  a 
similar  manner  and  the  measurement  of  AB  is  computed. 
When  this  is  done  the  instrument  is  carried  forward  to  B, 
where  the  fence-corners  are  located  and  the  notes  recorded  as 
at  A.  The  station  G is  next  selected,  and  the  angle  CBA  is 
measured  and  verified  as  before. 

Whenever  lines  of  the  traverse  are  incomplete  on  a page,  as 
AL , BM,  and  BG  on  page  2,  the  stations  to  which  they  run 
and  the  page  of  the  book  where  notes  around  those  stations 
are  to  be  found  are  recorded  as  shown.  A page  should  also  be 
assigned  to  a description  of  some  of  the  principal  stations,  so 
that  they  may  be  found  in  case  of  a resurvey.*  This  is  done  by 
giving  the  distances  from  the  station  to  two  objects,  such  as  a 
cut  on  a rock  or  a spike  driven  into  a tree;  lines  from  these 
reference  points  to  the  station  should  form  as  nearly  as  possi  • 
ble  a right  angle. 

Prob.  15.  Compute  the  area  of  the  polygon  AB . . . K from 
the  above  notes.  Also  compute  the  areas  tp  be  added  to  cr 
subtracted  from  it,  in  order  to  find  the  area  of  the  farm. 

Art.  16.  Office  Work. 

Office  work  embraces  computations  and  the  drawing  of 
maps.  The  method  of  computing  the  area  of  a polygon  has 
been  explained  in  Art.  6.  It  is,  however,  rarely  practicable  to 
have  the  lines  of  the  survey  coincide  with  the  boundaries  of 
the  field  or  farm,  and  hence  the  areas  of  the  trapezoids  between 
the  offsets  are  to  be  separately  computed  as  explained  in  Art. 
3,  and  these  are  added  to  or  subtracted  from  the  area  of  the 
polygon,  as  the  ease  may  require.  All  computations  should  be 
checked  so  that  the  results  may  be  relied  upon. 

In  order  to  facilitate  the  work  of  plotting  the  map  the  lati- 
tudes and  longitudes  of  the  principal  stations  are  often  com* 


OFFICE  WORK. 


55 


puted.  For  example,  in  Art.  6,  Fig.  10,  it  is. most  convenient 
to  take  tlie  point  A as  tlie  origin  of  coordinates.  The  latitude 
and  longitude  of  B are  then  the  same  as  the  latitude  and  longi- 
tude differences  of  AB.  For  the  station  C and  B , 

Lat.  C = 799.94  + 249.98  = 1049.92 
Long.  G = 0.00  + 438.07  = 433.07 

Lat  D = 1049.92  - 84.53  = 965.39 
Long.  B 2=  433.07  + 181.29  = 614.36 

and  in  like  manner  the  latitude  and  longitude  of  each  station 
is  found  from  those  of  the  preceding  station  by  simply  adding 
or  subtracting  the  adjusted  latitude  and  longitude  differences 
of  the  line. 

To  plot  the  field  to  a suitable  scale,  one  of  two  methods  is 
pursued  : the  sides  of  the  polygon  are  laid  off  in  succession  by 
the  angle  with  the  preceding  course,  and  the  length  of  the 
course;  or  each  corner  is  located  independent  of  all  the  others 
by  means  of  its  previously  computed  co-ordinates. 

In  plotting  by  the  first  method  the  angles  are  laid  off  either 
by  the  protractor,  or  by  their  natural  sines  or  tangents.  Be- 
fore using  the  protractor  the  azimuths  of  all  the  courses  with 
reference  to  any  one  of  them  are  computed.  The  direction  of 
this  course  is  drawn  and  the  protractor  is  placed  in  position 
upon  it  and  fastened;  all  the  azimuths  are  pricked  off  around 
the  edge  of  the  protractor  and  the  latter  is  removed.  The  di- 
rections of  all  the  courses  have  now  been  plotted  and  they  may 
be  transferred  to  any  part  of  the  paper  by  using  triangles. 
The  direction  of  any  course  as  AB  is  drawn  in  the  desired 
position  on  the  paper  and  its  length  measured  by  the  proper 
scale;  the  direction  of  BO  as  determined  by  the  protractor  is 
transferred  till  it  passes  through  B , and  the  position  of  station 
C found  by  measuring  on  this  line  the  length  of  BC.  In  like 
manner  all  the  courses  are  plotted  and  the  accuracy  of  the  work 
is  proved  if  the  point  A,  plotted  in  order  after  the  others, 
coincides  with  the  position  assumed  for  it  at  first. 

To  lay  off  an  angle  by  means  of  its  natural  sine  an  arc  is 
drawn  whose  radius  is  10  on  any  scale.  A chord  to  this  arc 
whose  length  is  the  sine  of  half  the  angle,  measured  with  a 


56 


LAND  SURVEYING. 


scale  twice  as  large  as  before,  will  subtend  tbe  angle  at  the 
center.  Thus  to  plot  the  angle  ABC  of  40°,  with  B as  a center, 
an  arc  is  drawn  with  a radius  of  10  to  the  scale  of,  say,  20  feet 
to  the  inch  ; with  the  intersection  of  this  arc  and  AB  as  a 
center  strike  an  arc  with  a radius  3.42  on  the  scale  of  10  feet 
to  the  inch,  cutting  the  first  arc  at  G,  then  ABC  is  the 
required  angle. 

To  plot  the  same  angle  by  using  its  tangent,  mark  a distance 
10  to  any  convenient  scale  from  B toward  A ; at  that  point 
erect  a perpendicular,  whose  length  is  8.39  to  the  same  scale, 
to  Gt  and  ABG  is  the  angle  desired. 

The  first  method  of  plotting  a map  has  the  merit  of  being 
easy  and  rapid,  but,  as  each  point  is  established  with  reference 
to  the  preceding  one,  any  error  in  the  location  of  a station  will 
affect  the  position  of  all  that  are  fixed  after  that  one,  and  it  is 
to  overcome  this  difficulty  that  the  method  by  co-ordinates  is 
used. 

After  the  coordinates  of  the  stations  have  been  computed  by 
taking  the  algebraic  sum  of  the  latitude  and  longitude  projec- 
tions of  the  preceding  courses,  the  origin  and  axes  of  coordi- 
nates are  plotted  upon  the  paper.  If  the  map  is  a large  one  the 
utmost  care  must  be  taken  to  make  the  angle  between  the  axes 
exactly  90°  ; the  right  angle  is  first  drawn  in  the  usual  way 
and  then  verified  by  measuring  the  hypothenuse  of  the  tri- 
angle as  large  as  the  limits  of  the  drawing  will  allow.  Paral- 
lel to  these  axes  lines  are  drawn  dividing  the  paper  into 
squares  100  feet,  200  feet,  or  1000  feet  on  a side,  according  to 
the  scale  of  the  drawing,  the  object  being  to  bring  every  point 
on  the  map  within  the  length  of  the  scale  from  two  of  these 
lines.  The  stations  may  now  be  located  by  measuring  their 
coordinates  from  the  nearest  parallels  and  the  accuracy  tested 
by  the  length  of  the  sides.  In  plotting  the  houses,  fences,  and 
brooks,  the  scale  is  placed  on  the  traverse-line  and  all  the  dis- 
tances along  its  length,  to  points  where  offsets  are  taken, 
are  measured  without  moving  it ; the  offsets  are  then  measured 
and  the  figures  completed. 

The  finished  map  should  contain  full  information  concerning 
the  date  of  survey,  scale  of  map,  names  of  owners  of  adjoining 


OFFICE  WORK. 


5? 


property,  and  of  tlie  surveyor  ; if  a portion  of  the  plan  has 
been  compiled  from  other  maps  that  fact  should  be  stated  and 
references  given.  The  title,  meridian  point,  and  border  are, 
in  a measure,  an  opportunity  for  the  exercise  of  artistic  skill 
on  the  part  of  the  draftsman,  but  legibility  and  simplicity 
must  not  be  sacrificed  for  ornament.  A title  of  Roman  letters, 
well  done,  always  presents  a good  appearance,  and  without 
other  decoration,  will  be  in  good  taste  on  maps  both  large  and 
small.  The  meridian  is  usually  represented  by  an  arrow  hav- 
ing the  head  at  the  north  end,  and  by  an  elongated  S at  the 
south  ; the  lines  should  be  very  light,  that  the  direction  may 
be  well  defined.  When  both  the  true  and  magnetic  meridians 
are  shown,  the  former  is  represented  by  a full  arrow  and  the 
latter  by  one  having  but  one  side  of  the  head  drawn.  The  ap- 
pearance of  the  border  is  sometimes  improved  by  geometrical 
figures  or  some  simple  ornament  in  the  corners,  but  a departure 
from  the  practice  of  using  simply  a light  line  on  the  inside  and 
a heavy  one  outside,  with  a space  between  them  as  wide  as  the 
heavy  line,  will  be  for  the  worse  oftener  than  for  the  better. 

Prob.  16.  Compute  the  coordinates  of  the  stations  for  Fig. 
23,  and  plot  the  map  of  the  farm  on  a scale  of  100  feet  to  one 
inch. 


Art.  17.  Random  Lines. 

A random  line  is  a line  run  out  in  order  to  find  a lost  corner, 
or  to  locate  a boundary  line  which  has  become  obliterated. 
For  example  in  Fig.  32,  let  A be  a given 
corner  and  let  \t  be  known  from  an  old  » 
record  that  a certain  line  AP  was  once 
established  having  a bearing  N 41°  30' 

W and  a length  of  32  chains.  No  traces 
of  this  line  or  of  the  corner  P are  now 
visible,  and  it  is  required,  if  possible,  to 
relocate  them.  Between  the  date  of  the 

old  survey  and  the  present  one  the  decli-  w A 

nation  of  the  needle  has  changed  several  Fig.  32. 

degrees,  perhaps,  and  the  first  duty  of  the  surveyor  is  to  con- 
sider this  question  carefully  and  ascertain  the  probable  amount 


58 


LAND  SURVEYING. 


of  change,  so  as  to  determine  the  present  probable  bearing  of 
the  line.  Suppose  that  the  result  of  this  inquiry  leads  to 
N 38°  L5'  W as  this  bearing. 

Starting  at  the  marked  corner  A the  surveyor  runs  a random 
line  AB  on  the  bearing  N 38°  15'  W,  and  measures  along  that 
line  a distance  of  32  chains,  or  2112  feet,  to  a point  B.  He 
then  proceeds  to  look  over  the  ground  on  both  sides  of  B for 
the  lost  corner,  which  is  described  in  the  old  record  as  a 
marked  tree,  a stump,  a pile  of  stones,  or  a monument.  If  it 
is  impossible  to  find  a trace  of  it  nothing  further  can  be  done 
I from  the  data  in  hand.  If,  however,  it  is  found  at  P,  a per- 
pendicular PE  is  dropped  upon  the  line  AB  and  its  length  is 
measured,  as  also  the  distance  BE.  The  distance  AE  is  thus 
known,  and  from  the  right  triangle  the  angle  EAP  can  be  com- 
puted and  the  present  magnetic  bearing  of  AP  be  determined. 
For  example  : Suppose  that  PE  is  found  to  be  37.4  feet, 

while  AE  is  2110.5  feet,  then 

P777  4- 

ta^P  = 3S=5iiOi =0'0m3’ 

whence  EAP  — 1°  01',  and  hence  the  present  magnetic  bearing 
of  AP  is  N.  39°  16'  W.  The  distance  AP  is 

Ollf)  K 

^LP  = :=  2110.8  feet, 

cos  1 01 

which  indicates,  if  the  present  work  is  accurate,  that  the  old 
survey  was  in  error  by  1.2  feet.  However,  it  is  a principle  of 
law  that  established  corners  and  monuments  must  control  re- 
surveys, and  hence  the  new  record  for  the  line  AP  is  N 39° 
16'  W 2110.8  feet 

Intermediate  points  on  the  line  AP  may  now  be  established 
by  starting  at  A and  running  it  out  with  the  new  bearing.  A 
quicker  way,  however,  is  to  lay  off  perpendiculars  from  the 
stakes  previously  set  on  the  line  AE,  marking  their  lengths 
proportional  to  the  distances  from  A.  For  instance,  if  it  be 
required  to  mark  a point  at  the  middle  of  AP,  the  perpendicu- 
lar to  be  erected  at  the  middle  oi  A E will  be  18.7  feet  in  length. 

Random  lines  are  also  frequently  used  to  find  the  bearing 
and  distance  between  two  points  which  are  not  intervisible, 


RESURVEYS. 


59 


?or  example,  let  G and  II  in  Pig.  33  be  two  sucb  points. 
Starting  at  G let  a line  GA  be  run  in  a direction  which  is  ap- 
3roximately  toward  II.  On  arriving  at 
A,  where  II  can  he  seen  let  AII  he  run. 

Suppose  that  GA  is  N 42°  15'  E,  714.5 
feet;  and  that  AH\s  N 1°  08'  W,  210.5 
feet.  It  is  required  to  find  the  length 
and  hearing  of  GII. 

For  this  purpose  the  length  of  each 
line  is  multiplied  by  the  sine  and  cosine 
of  its  hearing,  and  the  results  tabulated 
as  below.  The  principle  that  the  sum 
of  the  northings  equals  the  sum  of  the  southings,  and  the  sum 
of  the  eastings  equals  the  sum  of  the  westings  (Art.  7),  gives 
739.4  feet  for  the  southing  of  HG  and  480.4  feet  as  its  westing. 
Dividing  the  second  of  these  by  the  first  gives  the  tangent  of 


| Course.  Bearing.  Length.  Northing.  Southing.  Easting.  Westing. 

GA  N 42°  15'  E 714.5  528.9  480.4 

AH  N 1 08  W 210.5  210.5  4.2 

HG  (739.4)  (476.2) 

739.4  739.4  480.4  480.4 

the  angle  between  HG  and  the  meridian,  while  the  square 
root  of  the  sum  of  their  squares  is  the  length  of  IIG.  Thus 
the  bearing  of  HG  is  S 33°  01'  W,  and  that  of  GH  is  N 33°  01' 
E,  while  the  length  is  881.7.  This  length  can  also  be  found 
by  dividing  739.4  by  the  cosine  of  33°  01',  or  by  dividing  480.4 
by  the  sine  of  33°  01\ 


Prob.  17.  In  order  to  find  the  direction  and  distance  be- 
tween two  points  K and  L,  the  following  lines  are  run  : KA, 
S 87°  37'  W,  930.57  feet ; AB , West,  621.03  feet  ; BLy 
S 88°  15'  W,  82.78  feet.  Compute  the’  bearing  and  length  of 
&L,  and  locate  the  point  where  it  crosses  AB. 


Art.  18.  Restjrveys. 

When  several  lines  of  the  boundary  of  a farm  or  town  have 
become  obliterated  and  the  corners  lost,  it  is  often  necessary 
to  make  a resurvey  in  order  to  re-establish  them.  If  the  corners 


60 


LAND  SURVEYING. 


can  be  found  or  be  located  by  reliable  evidence  they  must  be 
accepted  as  correct  even  if  the  recorded  bearings  and  lengths 
of  the  lines  indicate  different  points.  It  sometimes  happens 
that  some  corners  can  be  found  while  others  cannot.  In  such 
cases  a series  of  random  lines  is  to  be  run  with  the  old  bear- 
ings, or  with  the  old  bearings  corrected  for  the  change  in 
declination  of  the  needle  between  the  two  dates. 


C 


As  an  example  let  the  records  in  an  old  deed  give  the  bear 
ings  and  lengths  of  three  lines  as  follows: 


Ab, 

N 60°  E, 

10  chains; 

be, 

N 45  E, 

4 chains; 

cd, 

S 45  E, 

8 chains. 

There  being  no  definite  data  at  hand  to  determine  the  change 
in  magnetic  declination  between  the  dates  of  the  two  surveys, 
the  lines  AB,  BC,  and  CD,  are  run  with  the  given  bearings 
and  distances  from  the  known  corner  A.  The  old  corners  b 
and  c cannot  be  found,  but  on  arriving  at  D the  old  corner  d is 
discovered  at  a point  distant  20.4  links  and*  S 12°  W from  B. 
It  is  required  to  locate  the  old  corners  b and  c. 

By  the  method  explained  in  Arts.  7 and  17,  the  bearings  and 
the  lengths  of  the  lines  DA  and  dA  may  be  computed.  These 
are : 

DA,  S 82°  47'  W,  17.29  chains; 

dA,  S 83  26  W,  17.22  chains. 

Now  the  error  Dd  between  the  two  corners  is  due  to  two 
causes  : first,  to  a constant  difference  in  the  magnetic  bear* 


RESURVEYS. 


61 


i„gs  of  the  two  surveys;  and  second,  to  a difference  in  the 
lengths  of  the  chains  used.  The  first  cause  swings  the  polygon 
AbcdA  around  the  point  A by  a small  angle.  The  second 
cause  alters  the  lengths  of  the  sides  in  a constant  ratio.  The 
difference  between  the  bearings  of  DA  and  (Ll  is  the  constant 
ancle,  while  the  ratio  of  the  lengths  of  these  lines  is  the  con- 
stant ratio.  To  find  the  bearings  of  the  old  lines,  therefore, 
each  of  the  given  bearings  is  to  be  corrected  by  the  amount 

83°  26'  ~ 82°  47'  = 0°  39', 

and  to  find  the  lengths  of  the  old  lines  each  of  the  given 
lengths  is  to  be  multiplied  by 

= 0.996. 

17.29 

All  of  this  reasoning  supposes  that  the  new  work  is  done 
with  such  precision  that  the  errors  in  chaining  must  be  re- 
garded  as  being  in  the  old  survey. 

Applying  these  corrections  the  adjusted  bearings  and  lengths 
of  the  old  lines  are 


Ab, 

N 

60°  39' 

E, 

9.96  chains; 

be, 

N 

45  39 

E, 

3.99  chains; 

cd, 

S 

44  21 

E, 

7.97  chains, 

and  with  these  new  data  the  lines  may  be  rerun  and  the  cor- 
ners b and  e be  located,  a check  on  the  field  work  being  that 
the  last  line  should  end  exactly  at  the  old  corner  d. 


It  is,  however,  not  difficult  to  compute  the  lengths  and 
bearings  of  Bb  and  Gc,  so  that  b and  e may  be  located  from  the 
points  B and  C.  The  principle  for  doing  this  is  that  the  poly- 
gons ABCDA  and  AbcdA  are  similar.  Thus  the  triangles 
ABb  and  ADd  are  similar;  hence  the  length  of  Bb  is 


Bb  - Dd  AD  - 


20.4  X 10 
17.29 


11.8  links. 


Also  the  angle  ABb  equals  the  angle  ADd,  or  70°  47' ; hence 
the  bearing  of  Bb  is  S 10°  47'  E.  In  like  manner,  the  triangle 
ACc  being  similar  to  ADd,  the  length  and  bearing  of  Gc 
can  be  found,  the  length  and  bearing  of  AC  being  first  com- 
puted. The  distance  Gc  is  16.4  links,  and  its  bearing  is 


62 


LAND  SURVEYING. 


S 15°  03'  E.  The  lines  Bb  and  Cc  are  now  run  from  B and  0, 
and  thus  the  most  probable  location  of  the  old  corners  b and  c 
is  made. 

Prob.  18.  The  record  of  an  old  survey  reads  as  follows  : 
Commencing  at  a post  marked  No.  5 and  running  N 62°  E, 

14.00  chains,  to  a stake  marked  A;  thence  running  N 43£  E, 

8.00  chains,  to  a stake  B;  thence  N 5&  W,  12.00  chains,  to  a 
stake  B;  thence  N 72£°  E,  10.25  chains,  to  a stake  I);  thence 
S 12°  W,  6.43  chains,  to  a stone  marked  No.  3.  On  rerunning 
the  lines  the  end  of  the  last  one,  instead  of  being  at  the  stone 
No.  3,  was  0.62  chains  due  East  from  it.  Find  the  adjusted 
bearings  and  lengths  of  the  old  lines;  also  find  the  distance  and 
direction  from  each  station  of  the  new  survey  to  the  corre- 
sponding one  of  the  old  survey. 

Art.  19.  Traversing. 

The  term  traverse,  which  was  originally  associated  with 
navigation,  is  in  common  use  by  surveyors  to  define  a series 
of  lines  whose  lengths  and  relative  directions  are  known.  For 
example  in  Fig.  23  the  lines  T8,  SB,  BP,  constitute  a trav- 
erse run  for  the  purpose  of  locating  a brook.  Traversing  is 
particularly  applicable  to  the  survey  of  long  and  circuitous 
routes  through  territory  presenting  natural  obstructions  to 
long  sights.  It  is  almost  univerally  adopted  in  filling  in  the 
interior  of  maps  which  are  based  upon  a system  of  -triangula- 
tion. As  examples  of  traversing  may  be  mentioned  the  survey 
of  highways  and  railroads,  river  banks,  shores  of  lakes,  and 
property  boundaries.  In  the  United  States  Government  sur- 
veys, when  the  traverse  is  run  to  mark  the  division  between 
private  estates  and  a body  of  water  retained  as  public  property 
it  is  called  a Meander  Line. 

The  most  approved  method  of  running  a traverse  is  that  in 
which  the  graduated  plate,  or  limb,  of  the  transit  is  so  set  at 
each  station  that  the  azimuth  of  each  line  there  observed  can 
be  directly  read.  If  the  survey  is  made  in  a locality  where  no 
system  of  latitudes  and  longitudes  has  been  established,  the 
magnetic  meridian  may  be  taken  as  the  meridian  of  the  azi- 
muths. At  the  first  station  the  vernier  is  set  at  zero  and  by 


TRAVERSING. 


63 


means  of  the  lower  motion  the  instrument  is  turned  so  that  the 
north  end  of  the  needle  points  to  the  JSf  on  the  compass  limb. 
The  lower  plate  being  then  clamped  the  upper  one  is  un- 
damped; now  if  a sight  be  taken  at  any  object  the  reading  on 
the  vernier  will  be  the  azimuth  corresponding  to  the  bearing 
of  that  object.  The  last  sight  and  reading  taken  at  the  first 
station  is  toward  the  second  station  of  the  traverse  line.  The 
instrument  is,tlien  placed  over  the  second  station  and  the  ver- 
nier set  at  the  back  azimuth  of  the  first  station ; the  azimuth 
of  any  line  from  the  second  station  will  now  correspond  with 
its  bearing  as  before.  The  readings  of  the  needle  are  recorded 
as  a rough  check  on  the  azimuths,  with  which  they  should 
agree  to  the  nearest  eighth  of  a degree. 

For  example,  at  the  station  A let  the  bearing  of  AB  be 
N 74°  15'  E,  and  let  its  azimuth  be  74°  15'.  On  placing  the 
instrument  at  B , the  vernier  is  set  at  254°  15',  a sight  taken  on 
Af  and  the  lower  plate  clamped.  The  azimuth  of  BO  being 
113°  02',  the  vernier  is  set  at  323°  02'  on  arriving  at  0 and  the 
Ij  mb  placed  in  proper  position  by  sighting  back  to  B.  The 
it  descope  is  not  reversed  during  any  part  of  the  work.  At 
exch  of  the  stations  sights  may  be  taken  to  surrounding  ob- 
it ;cts,  and  if  the  distance  to  an  object  is  measured  this  together 
with  its  azimuth  locates  it  with  respect  to  the  station. 


Bearing.  * 

Azimuth. 

Distance. 

Object  Sighted. 

Notes 

at  Station 

B 

S 74°  15'  W 

234°  15' 

528.3 

Station  A 

325  42 

250. 

Large  pine  tree 

196  24 

NE  corner  of  John  Doe’s  House 

194  10 

SE  corner  of  J. Doe’s  same  House 

S 37*  00'  E 

143  02 

490.7 

Station  C 

Notes 

at  Station 

C 

,N  37°  05'  W 

323°  02' 

490.7 

Station  B 

280  13 

NE  corner  of  John  Doe’s  House 

276  15 

SE  corner  of  J.Doe’s  same  House 

104  07 

98  5 

Fence  corner 

S 42°  45'  E 

137  15 

504.6 

Station  D 

The  field  notes,  if  offsets  are  taken  from  the  traverse  lines 
are  best  kept  as  in  Figs.  24-31,  the  bearing  of  a line  being 
written  upon  one  side  of  it  and  the  azimuth  upon  the  other  side. 


G4 


LAND  SURVEYING. 


If  no  offsets  are  taken  a form  like  tliat  given  above  may  be 
used.  It  is  seen  that  the  large  pine  tree  is  located  by  azimuth 
and  distance,  at  station  B,  as  also  is  the  fence  corner  at  station 
G.  The  house  of  John  Doe,  however,  is  located  by  azimuths 
taken  from  both  B and  G,  the  line  BG  forming  a base  by  which 
its  distance  from  either  end  can  be  computed. 

It  is  always  desirable  that  a traverse  should  have  a check 
upon  its  accuracy.  In  a closed  traverse  like  that  around  the 
boundaries  of  a farm  this  is  obtained,  since  the  sum  of  the 
northings  must  equal  the  sum  of  the  southings,  and  the  sum 
of  the  eastings  that  of  the  westings.  In  Fig.  23,  the  traverse 
GNOPQG,  which  begins  at  G and  ends  at  Gy  is  checked  in  the 
field  on  arriving  at  G , for  the  azimuth  of  GH  must  agree  with 
that  previously  obtained  ; also  in  computation  the  differences 
of  latitude  and  longitude  between  G and  G must  agree  with 
those  obtained  from  the  main  polygon. 

It  should  be  remarked  that  the  object  of  taking  the  bearings 
is  merely  to  check  gross  errors  in  the  azimuths  during  the 
progress  of  the  field  work,  and  that  an  experienced  engineer 
will  usually  prefer  to  take  but  few  readings  of  the  needle.  If 
a true  meridian  has  been  established  in  the  neighborhood  of 
the  survey  the  azimuths  should  be  reckoned  from  it  instead  of 
from  the  magnetic  meridian.  „ 

Frob.  19.  Compute  from  the  above  notes  the  length  of  the 
west  side  of  John  Doe’s  house.  Obtain  the  same  distance 
without  computation  by  plotting  the  notes. 

Art.  20.  United  States  Public  Land  Surveys. 

The  system  adopted  by  the  United  States  Government  on 
May  20,  1785,  for  the  survey  of  the  public  land  which  had 
been  acquired  from  time  to  time,  consists  in  dividing  it  into 
squares,  called  townships,  six  miles  on  a side,  by  meridians 
and  east  and  west  lines.  A north  and  south  row  of  townships 
is  called  a range.  The  townships  are  divided  into  square 
miles,  called  sections,  which  are  subdivided  into  half  and 
quarter  sections. 

The  work  of  surveying  the  government  land  is  begun  by 


UNITED  STATES  PUBLIC  LAND  SURVEYS. 


65 


carefully  running  a north  and  south  line,  called  the  principal 
meridian,  and  an  east  and  west  line  called  the  standard  parallel. 
The  standard  parallels  are  24  miles  or  BO  miles  apart,  according 
as  they  are  above  or  below  35°  north  latitude,  and  the  princi- 
pal meridians  are  at  long  intervals — 100  miles  or  more.  On 
these  lines  every  mile  is  marked  by  a stake  or  monument  and 
called  a section  corner  ; every  sixth  section  corner  is  called  a 
township  corner  and  is  differently  marked. 

On  the  standard  parallel  the  township  corners  are  next 
marked  ; from  each  of  these  corners  meridians  are  run  to  in- 
tersect the  standard  parallel  next  north.  Owing  to  the  con- 
vergence of  meridians  toward  the  pole,  the  points  of  their  in- 
tersections with  the  standard  parallel  will  not  be  at  the 
township  corners,  but  a little  nearer  the  principal  meridian  ; 
as  the  full  six  miles  have  been  measured  on  the  standard  par- 
allels, the  convergence  is  corrected  at  each  of  those  iines. 

At  each  of  the  township  corners  on  the  principal  meridian, 
east  and  west  lines  are  run  intersecting  the  meridians  through 
section  corners  ; on  these  parallels  the  section  corners  are 
marked  one  mile  apart  for  five  miles,  the  remaining  section 

being  one  mile  less  the  amount  of  meridinal  convergence  for 

% 

the  distance  to  the  standard  parallel  next  south. 

The  meridians  through  the  section  corners  are  run  for  five 
miles,  then  from  the  points  where  they  intersect  the  fifth  east 
and  west  section  lines,  oblique  lines  are  run  to  the  points  pre 
viously  established  on  the  northern  boundary  of  the  township  * 
when,  however,  the  northern  boundary  of  the  township  is  one 
of  the  standard  parallels,  the  section  meridians  are  run  directly 
the  full  six  miles  instead  of  deflecting  at  the  fifth  east  and 
west  line. 

The  convergence  of  the  meridians  is  given,  very  nearly,  by 
the  following  rules  of  geodesy  : 

The  angular  meridional  convergence  equals  the  difference  in 
longitude  into  the  sine  of  the  latitude. 

The  linear  convergence  equals  the  distance  along  the  me- 
ridian into  the  sine  of  the  angular  meridional  convergence. 

The  townships  are  divided  into  36  sections,  numbered  from 


66 


LAND  SURVEYING. 


1 to  36,  as  shown  in  Fig.  35.  The  sections  themselves  are 
subdivided  and  designated  as  in  Fig.  36  ; a represents  the  va- 
rious ways  of  dividing  an  entire 
section,  and  b shows  the  method 
N when  a portion  of  the  section  is 
| obstructed  by  water.  In  cases 
Jr_  of  this  kind  it  is  usual  to  add  to 
j an  adjacent  lot  the  salable  part 
| of  the  obstructed  quarter  section, 
s and  to  state  the  total  number  of 
acres  in  both;  but  when  only  a 
small  portion  of  the  quarter 
section  is  unsalable  it  retains  its  own  name,  is  called  fractional, 
and  the  number  of  acres  in  it  are  given. 


G 

5 

4 

3 

2 

1 

7 

8 

9 

10 

11 

12 

18 

17 

16 

15 

14 

13 

19 

20 

21 

22 

23 

24 

30 

29 

28 

27 

26 

25 

31 

32 

33  | 

34 

35 

36 

Fig.  35. 


North  % 

N.W.  % 
of 

S.W.  \i 

E.  \i 
of 

S.  E.  H 

s.w.  yi 

of 

s.w.  H 

S.W.  \i 

Fig.  36. 

The  methods  of  running  the  principal  meridians  and  stand- 
ard parallels  are  founded  on  the  science  of  geodesy.  The  rules 
governing  the  running  of  township  and  section  lines  may  be 
found  in  “Instructions  to  the  Surveyors  General  of  Public 
Lands,  ” issued  by  the  Land  Office  of  the  Interior  Department, 
Washington,  D.  C.  The  principles  of  this  chapter  and  the  last 
are,  however,  directly  applicable  to  the  surveying  and  map- 
ping of  townships,  sections,  and  their  subdivisions. 


Prob.  20.  Compute  the  length  of  the  northern  and  southern 
boundaries  of  a township  in  latitude  46°  30',  the  southern 
boundary  being  18  miles  north  of  a standard  parallel. 


THE  LEVEL, 


67 


CHAPTER  III. 

LEVELING  AND  TRIANGULATION. 

Art.  21.  Tiie  Level. 

The  Engineer’s  Level  consists  of  a line  of  sight  parallel  to  a 
spirit  level  and  perpendicular  to  a vertical  axis.  The  line  of 
sight  is  fixed  in  a telescope  by  cross-hairs  as  in  the  transit. 
The  spirit  level  is  attached  to  the  under  side  of  the  telescope 
and  is  protected  except  on  top  by  a metal  tube.  The  telescope 
is  supported  on  vertical  forks,  called  Ys  (from  which  fact  the 
instrument  is  called  the  Y level),  and  is  clamped  to  them  by 
collars  which  may  be  raised,  allowing  the  telescope  to  be 
turned  on  its  axis  or  taken  out  entirely.  The  Ys,  which  may 
be  lengthened  or  shortened  by  screws  for  the  purpose,  are 
fastened  to  a horizontal  bar  which  is  rigidly  attached  to  the 
vertical  axis.  The  instrument  is  provided  with  leveling 
screws  and  mounted  upon  a tripod. 

The  Dumpy  Level  differs  from  the  ordinary  form  in  having 
the  telescope  firmly  fixed  on  the  horizontal  bar  so  it  cannot  be 
turned  either  on  its  axis  or  end  for  end.  This  level  is  superior 
to  the  Y type  in  every  point  of  difference,  being  less  costly, 
lighter,  and  more  permanent  in  its  adjustment.  The  superior 
ity  claimed  for  the  Y level  is  the  ease  of  adjustment  by  means 
of  its  movable  telescope,  but  if  such  an  advantage  exists  it  is 
extremely  slight. 

The  parts  of  the  level  of  most  importance  are  the  telescope 
and  the  bubble.  The  character  of  the  work  to  be  done  will 
determine  whether  or  not  magnifying  power  in  the  telescope  is 
more  desirable  than  illumination  of  the  field  of  view  and  what 
was  said  on  this  subject  in  connection  with  the  transit  applies 
as  well  to  the  level.  The  upper  part  of  the  inside  surface  of 
the  bubble  tube  is  carefully  ground  in  the  form  of  a longitu- 
dinal circular  curve,  and  upon  the  radius  of  this  curve  depends 
what  is  known  as  the  sensitiveness  of  the  level.  If  the  radius 
of  curvature  of  the  bubble  is  large  it  will  be  very  sensitive ) 


08 


LEVELING  AND  TRIANGULATION. 


that  is,  a slight  vertical  displacement  of  the  telescope  will 
cause  a considerable  motion  of  the  bubble.  If  the  radius  of 
curvature  is  short  the  bubble  is  not  sensitive.  A very  sensi- 
tive bubble  is  not  desirable  since  much  time  will  then  be  lost 
in  leveling  the  instrument. 

The  level  rod  is  a graduated  scale  for  measuring  the  vertical 
distance  between  the  horizontal  plane  through  the  line  of  sight 
and  that  through  the  point  upon  which  the  rod  is  held.  Tar- 
get rods  are  used  in  precise,  work,  and  self-reading  rods  in 
cases  where  elevations  need  to  be  determined  only  to  tenths  of 
a foot.  The  target  rod  has  a vernier  on  its  movable  target  by 
which  readings  to  the  thousandth  of  a foot  are  taken  by  the 
rodman  ; the  New  York  rod,  the  Boston  rod,  and  the  Philadel- 
phia rod  are  the  most  common  forms  in  use.  Self-reading  rods 
have  figures  and  graduations  distinct  enough  to  be  read  by  the 
leveler  as  he  sights  through  the  telescope.  A self-reading  rod 
is  divided  into  tenths  of  a foot,  but  if  the  figures  are  properly 
made  readings  to  hundredths  of  a foot  can  easily  be  taken  ; the 
numbers  marking  the  tenths  should  be  0.06  feet  long  and  so 
placed  that  half  the  length  is  above  and  half  below  the  line. 
The  numbers  marking  the  feet  are  0.10  feet  long,  and  each  is 
bisected  by  the  foot-mark. 

Prob.  21.  Sketch  a part  of  a target  rod  showing  a vernier 
reading  5.027  feet.  Sketch  a self-reading  rod  according  to  the 
above  directions. 

Art.  22.  Adjustments  of  a Level. 

The  adjustment  of  an  instrument  consists  in  bringing  the 
various  parts  into  their  proper  relative’positions  so  that  all  4ke 
geometrical  conditions  necessary  for  good  work  may  be 
observed.  When  an  instrument  is  received  from  the  maker  it 
should  be  in  perfect  adjustment,  and  with  proper  care  it  will 
remain  so  for  a long  time.  It  should,  however,  be  examined 
at  frequent  intervals,  and  if  found  out  of  adjustment  at  any 
time,  should  be  at  once  put  into  proper  condition.  The  fol- 
lowing description  of  the  adjustments  of  the  Y level  follows  the 
order  in  which  they  should  be  made. 


ADJUSTMENTS  OF  A LEVEL. 


69 


Parallax. — This  is  an  improper  condition  of  focusing  due  to 
the  fact  that  the  image  does  not  fall  in  the  plane  of  the  cross- 
hairs. To  ascertain  if  it  exists,  direct  the  telescope  upon  the 
sky  and  focus  the  eyepiece  so  that  the  cross-hairs  are  perfectly 
distinct.  Then  turn  the  telescope  upon  the  object  which  is  to 
be  observed,  and. focus  the  object  glass  until  the  image  is  per- 
fectly distinct.  Move  the  eye  from  side  to  side  and  note 
whether  there  is  any  apparent  movement  of  the  cross-hairs  and 
image.  If  any  is  seen  the  two  operations  are  to  be  repeated 
until  all  parallax  is ‘removed.  This  adjustment  depends  upon 
the  eye  of  the  observer,  and  when  made  for  one  person  may 
not  be  correct  for  another. 

Collimation. — The  line  of  sight,  or  collimation,  should  not 
deviate  from  the  optical  axis  of  the  telescope.  To  ascertain  if 
an  error  in  collimation  exists,  loosen  the  collars  on  the  Y’s  and 
focus  the  telescope  upon  a distant  object.  Slowly  revolve  the 
telescope  in  the  Y’s  and  note  whether  the  intersection  of  the 
cross-hairs  remains  on  the  same  point.  If  the  horizontal  hair 
deviates  from  the  point  adjust  it  by  moving  it  over  half  the 
apparent  error,  by  means  of  the  capstan  screws  on  the  top  and 
bottom  of  the  telescope.  If  the  vertical  hair  deviates  adjust  it 
by  moving  it  over  half  the  apparent  error  by  means  of  the  cap- 
stan screws  on  the  sides  of  the  telescope.  The  instrument  is, 
of  course,  to  be  clamped  while  making  this  adjustment,  but  it 
need  not  be  leveled. 

The  Attached  Bubble. — The  level  bubble  attached  to  the 
telescope  must  be  parallel  to  the  line  of  sight.  To  ascertain  if 
this  is  the  case,  span  the  collars,  carefully  level  the  instrument 
and  clamp  it;  lift  the  telescope  out  of  the  Y^’s,  turn  it  end  for 
end,  and  replace  it.  If  the  bubble  does  not  settle  in  the 
middle  turn  the  screws  above  and  below  one  end  of  the  bubble- 
tube  so  as  to  bring  the  bubble  half  way  back.  Next  see  if  the 
bubble  is  in  the  same  plane  as  the  telescope  by  slowly  revolv- 
ing the  latter  in  the  Y’s  and  noting  whether  the  bubble  runs 
away  from  the  middle;  if  it  does  correct  half  the  apparent 
error  by  the  screws  on  the  sides  of  the  other  end  of  the  bubble- 
tube.  Repeat  these  operations  until  perfect  adjustment  is 
secured. 


70 


LEVELING  AND  TRIANGU LATION. 


Tlie  Horizontal  Bar. — The  telescope  and  level-bubble  should 
be  parallel  to  the  horizontal  bar  supporting  the  Y’s,  or  perpen- 
dicular to  the  vertical  axis  of  the  instrument.  To  ascertain  if 
this  is  the  case  after  the  preceding  adjustments  have  been 
made,  level  the  instrument  and  revolve  it  180  degrees  on  the 
vertical  axis.  If  the  bubble  runs  toward  one  end,  the  Y on 
that  end  is  too  high,  and  the  screws  at  the  end  of  the  horizontal 
bar  are  moved  so  as  to  correct  one  half  of  the  apparent  error. 
Then  repeat  the  operation  until  the  bubble  remains  in  the 
middle  of  the  scale  for  all  positions  of  the  telescope. 

In  adjusting  an  instrument  great  care  must  be  taken  not  to 
turn  the  screws  too  tight,  as  by  so  doing  the  threads  soon  be- 
come injured.  No  student  or  beginner  should  be  allowed  to 
adjust  a level  or  transit  until  he  has  become  well  acquainted 
with  all  its  parts  by  actual  use.  The  parallax  adjustment, 
however,  is  an  exception,  since  this  varies  for  different  eyes, 
and  each  student  should  see  that  this  is  made  every  time  he 
uses  the  instrument. 

The  dumpy  level  cannot  be  adjusted  by  the  above  methods 
since  the  horizontal  bar  and  telescope  are  rigidly  connected. 
Both  the  bubble  and  the  horizontal  cross-hair  are,  however, 
movable.  It  is  necessary,  (a)  that  the  bubble  should  be  per- 
pendicular to  the  vertical  axis  and  (b)  that  the  line  of  sight 
should  be  parallel  to  the  bubble.  The  adjustment  (a)  is  made 
exactly  like  that  above  described  for  the  horizontal  bar  of  the 
Y level.  The  adjustment  ( b ) is  made  by  the  peg  method  of 
Art.  26,  except  that  the  horizontal  cross-hair  is  moved  instead 
of  the  bubble. 

Prob.  22.  Give  the  reasons  for  each  of  the  adjustments  of 
the  Y level. 


Art.  23.  Comparison  of  Levels. 

In  buying  an  instrument  it  is  desirable  that  the  surveyor 
should  be  able  to  make  such  an  examination  as  will  indicate 
whether  it  is  a good  one  of  its  class  or  whether  it  is  the  kind 
that  he  needs.  The  following  tests,  which  are  useful  in  addi- 
tion to  those  of  the  last  article,  will  be  found  valuable  in 


COMPARISON  OF  LEVELS. 


71 


selecting  an  instrument,  or  in  comparing  one  with  another.  In 
making  them  the  instrument  should  be  in  good  adjustment. 

Magnifying  Power.— The  magnifying  power  of  a telescope 
may  be  obtained  by  dividing  the  focal  length  of  the  object 
glass  by  that  of  the  eyepiece.  As  these  however,  cannot  be 
closely  measured  the  following  method  is  usually  preferable: 
Place  a rod,  on  which  the  divisions  are  very  plainly  marked, 
about  25  yards  from  the  instrument  and  focus  the  telescope 
upon  it.  Turn  the  line  of  sight  slightly  away  from  the  rod 
and  focus  the  other  eye  upon  it.  Slowly  turn  the  telescope 
again  toward  the  rod,  when  the  small  image  as  seen  by  that 
eye  will  appear  projected  upon  the  larger  one  seen  through 
the  telescope.  If,  for  instance,  100  divisions  seen  by  the 
naked  eye  appear  to  cover  5 divisions  seen  by  the  other  eye 
through  the  telescope,  then  the  magnifying  power  is  100  -s-  5 
— - 20.  A high  magnifying  power  implies  a small  field  of  view 
and  hence  is  not  desirable.  For  a surveyor’s  transit  or  level  a 
magnifying  power  of  from  15  to  20  is  sufficient;  for  an  engi- 
neer’s transit  it  should  be  from  20  to  25,  and  for  an  engineer’s 
level  perhaps  from  25  to  30. 

Spherical  Aberration. — This  is  a defect  caused  by  combin- 
ing lenses  of  different  curvatures,  so  that  objects  on  the  sides 
of  the  field  of  view  are  seen  less  distinctly  than  those  in  the 
center.  To  test  the  object  glass  for  this  defect,  cover  the  outer 
edge  with  an  annular  ring  of  paper  and  focus  upon  a distant 
object;  then  remove  the  ring  and  cover  the  central  part  of  the 
glass;  if  no  change  of  focus  is  needed  the  glass  has  no  spheri- 
cal aberration.  To  test  the  eyepiece,  sight  to  a heavy  black 
line  drawn  on  white  paper  and  held  near  the  side  of  the  field 
of  view;  if  it  appears  perfectly  straight  the  eye  glass  is  a good 
one. 

Chromatic  Aberration.— This  is  a defect  caused  by  com- 
bining lenses-  of  improper  varieties  of  glass  so  that  yellow  or 
purple  colors  appear  on  the  edges  of  the  field.  To  test  a tele- 
scope for  this  defect,  focus  it  upon  a bright  distant  object  and 
slowly  move  the  object  glass  out  and  in;  if  no  colors  are 
observed  around  the  edges  of  the  field  of  view  the  telescope  is 
free  from  this  defect. 


72 


LEVELING  AND  TUI  ANGULATION. 


Definition. — The  ability  to  show  images  with  sharp,  clear 
outlines  is  a valuable  quality  in  a telescope.  It  may  be  tested 
by  comparing  the  distinctness  of  the  image  with  that  of  the 
object  as  seen  by  the  eye  at  such  a distance  that  it  will  seem 
the  same  in  size  as  the  image.  Ordinary  print  when  read  by 
the  eye  and  through  the  glass  with  equal  ease  should  appear 
equally  distinct. 

Size  of  Field. — The  angular  diameter  of  the  field  of  view  is 
usually  about  one  degree.  The  value  for  any  telescope  may  be 
closely  obtained  by  laying  off  a distance  of  57.3  feet  from  the 
object  glass,  placing  two  pins  in  the  ground  at  the  extreme  sides 
of  the  field,  and  measuring  the  distance  between  them  in  feet; 
this  will  be  the  size  of  the  field  of  view  in  degrees.  (Art.  2.) 


Sensitiveness  of  Bubble. — For  very  fine  work  the  radius  of 
curvature  of  a level  bubble  should  be  about  100  feet,  for  ordinary 
good  work  50  feet  is  preferable,  and  for  common  work  25  feet 
will  do.  To  determine  this  radius  let  the  instrument  be  set  up 
and  leveled,  so  that  two  screws  will  be-in  the  line  of  sight  to 
a target  rod  placed  100  feet  or 
more  away.  Let  one  end  of  the 
bubble  be  made  to  coincide  with 
one  of  the  division  marks  at  a and 
a reading  be  taken  on  the  rod  at  A. 
Then  by  the  two  screws  let  the  tel- 
escope be  raised  in  a vertical  plane 
until  the  end  of  the  bubble  reaches 
the  next  division  at  b,  when  a 
second  reading  is  taken  on  the  rod  at  B . Now,  if  B be  the 
radius  of  the  level  bubble  and  D the  distance  from  the  instru- 
ment to  the  rod,  B:  Bwab:  AB  very  nearly.  The  distance 
AB  is  the  difference  of  the  readings  on  the  rod,  while  ab  is  the 
length  of  one  space  of  the  bubble  scale ; thus  B is  known.  For 
example,  let  the  rod  be  150  feet  from  the  instrument,  the  two 
rod  readings  be  3.704  and  3.745  feet,  and  the  bubble  scale  have 
8 spaces  in  one  inch,  one  space  thus  being  of  a foot  long, 
Then 

D X ah  150 


Fig.  37. 


B = 


AB  0.041  X 96 


= 38.1  feet, 


LEVELING. 


73 


wliicli  is  the  radius  of  the  level  bubble.  The  operation  should 
now  be  repeated  using  a different  distance  D , and  the  mean  of 
several  results  be  taken  as  a final  value. 

Prob.  23.  A level  bubble  has  a radius  of  125  feet  and  its 
scale  has  10  spaces  in  an  inch.  Wliat  error  in  leveling  will 
result  at  a distance  of  250  feet  if  the  bubble  is  1^  spaces  out  of 
level  ? 


Art.  24.  Leveling. 

A Level  Surface  is  that  of  a fluid  at  rest,  and  a Level  Line  is 
the  intersection  of  such  a surface  with  a vertical  plane.  The 
line  of  sight  through  the  telescope  of  a properly  leveled  and 
adjusted  leveling  instrument,  when  revolved  around  the  verti- 
cal axis,  generates  a plane  which,  for  short  distances,  practi- 
t ally  coincides  with  the  level  surface  through  the  instrument. 


The  amount  of  deviation  between  the  two  surfaces,  due  to  the 
curvature  of  the  earth  and  to  refraction,  varies  as  the  square 
of  the  horizontal  distance  from  the  instrument  and  at  one  mile 
is  about  .57  feet. 

The  field  work  of  leveling  consists  in  finding  the  relative 
elevations  of  two  or  more  points.  The  elevations  are  referred 
to  an  assumed  surface  called  the  Datum  Plane,  or  simply 
Datum,  which  is  so  selected  that  all' points  whose  elevations 
'are  required  shall  be  above  it.  A mean  sea  level  is  frequently 
taken  as  the  datum  plane.  A Bench  Mark  is  a monument, 
rock  or  other  permanent  object  whose  elevation  above  the 
datum  has  been  determined.  The  method  of  carrying  on  the 
field  work  can  best  be  explained  by  Fig.  38.  The  line  MN  rep- 
resents the  datum  plane;  a is  a bench  mark  whose  elevation 
is  known;  b,  c,  d,  e,  f,  are  points  whose  elevations  are  desired; 


74 


LEVELING  AND  TRIANGULATION. 


A,  B,  and  C are  tlie  successive  positions  of  tlie  instrument. 
The  positions  of  the  rod  are  indicated  by  the  vertical  lines  and 
the  lines  of  sight  by  the  horizontal  dotted  ones.  The  instru- 
ment is  leveled  at  A and  the  reading  al,  on  *the  bench  mark  at 

a,  is  taken ; this  is  called  a Back  Sight  and  is  added  to  the  eleva- 
tion Ma,  to  get  the  Height  of  Instrument.  The  rod  readings  at 

b,  c,  and  d,  subtracted  from  the  height  of  instrument  will  give 
the  elevations  of  those  points  above  the  datum  MN ; such  read- 
ings are  called  Fore  Sights.  If  the  distance  Ad  is  as  far  as  can 
be  seen,  the  rod  is  kept  at  d,  which  is  called  a Turning  Point ; the 
instrument  is  carried  forward  to  B,  and  the  back  sight  dn  is 
taken ; the  new  height  of  instrument  is  then  Pd  + dn,  and  fore 
sights  at  e and  / are  taken  to  determine  the  elevations  of  the 
stations  e and/.  The  instrument  may  then  be  carried  forward 
to  C and  the  elevations  of  g,  li,  and  k determined  in  a similar  man- 
ner. If  the  instrument  is  always  set  midway  between  the  turn- 
ing points,  the  errors  in  rod  readings,  due  to  the  non-adjust- 
ment of  the  instrument  and  to  the  curvature  of  the  earth,  will 
be  confined  to  the  intermediate  points  as  b,  c,  and  e\  this  fact 
should  always  be  remembered  as  upon  it  depends,  in  a great 
measure,  the  accuracy  of  the  work.  The  turning  points  aie 
not  necessarily  taken  at  places  whose  elevation  is  desired,  but 
may  be  at  any  convenient  location,  either  on  or  off  the  lines; 
they  should  be  so  selected  that  an  unobstructed  view  of  the  rod 
may  be  had  from  any  probable  position  which  may  be  selected 
as  the  next  place  for  the  instrument,  and  be  upon  firm  objects 
which  cannot  be  readily  disturbed  while  the  instrument  is  be- 
ing carried  forward. 

The  field  notes  are  kept  as  shown  below;  they  are  usually  on 
the  left-hand  page  of  the  note  book  while  the  opposite  page  is 
devoted  to  remarks.  The  first  column  gives  the  name  or  num- 
ber of  the  point  where  the  rod  is  placed;  such  a point  is  called 
a Station.  If  the  stations  are  in  a continuous  line,  as  along  the 
middle  of  a road,  the  distances  between  them  are  given  in  the 
second  column.  The  back  sights  are  given  in  the  nextcoluimi; 
then  the  height  of  instrument,  foresight,  and  elevation,  in  the 
order  named.  This  arrangement  will  be  found  most  conven- 
ient in  making  the  additions,  for  the  height  of  instrument  and 


CONTOURS  ANT)  PROFILES. 


75 


tlie  subtractions  for  the  elevations.  It  is  seen  that  the  rod  is 
read  to  thousandths  of  a foot  on  the  bench  marks  and  turning 
points  and  to  hundredths  of  a foot  on  the  other  points.  In 
work  of  less  precision  than  that  in  towns  and  cities  the  rod 


Station 

Dist. 

B.S. 

H.I. 

F.S. 

Eleva. 

Remarks. 

a 

0 

6.320 

590.99) 

584.674 

Bench  mark  on  monu- 

b 

150 

2.12 

588.87 

[ment  No.  51. 

c 

200 

6.38 

584.61 

TP  A 

3.561 

584.243 

10.312 

580.682 

On  rock  50  ft.  N.E  of  c 

e 

280 

1.20 

583.04 

T.P.  / 

400 

10.617 

594.317 

0.543 

583.700 

On  rock. 

g 

475 

5.82 

588.50 

h 

500 

4.16 

590.16 

lc 

584 

3.245 

591.072 

B.M.on  stump  oak  tree 

readings  are  frequently  taken  only  to  hundredths  on  the  benches 
and  turning  points  and  to  tenths  on  the  others.  The  final  ele- 
vation of  the  bench  mark  k may  be  checked  thus: 


584.674+20.498  - 14.100  = 591,072 

in  which  20.498  is  the  sum  of  the  back  sights  on  the  benches 
and  turning-points  and  14.100  is  the  sum  of  the  fore  sights  on 
such  points.  (Art.  9.) 

When  levels  are  run  merely  to  find  the  difference  in  elevation 
of  two  points  a and  k (Fig.  38)  the  column  of  distances  is  not 
needed  in  the  notes,  and  there  are  no  intermediate  stations  b,  c, 
e,  g,  h.  It  is  well,  even  in  such  cases,  to  fill  out  the  column  of 
height  of  instrument  in  the  field,  and  to  check  the  final  result 
in  the  manner  indicated  above.  The  main  note  book  is  always 
kept  by  the  leveler,  but  the  rodman  should  also  keep  a book 
in  which  he  records  all  readings  on  benches  and  turning  points, 
finding  their  elevations  and  the  heights  of  instrument  so  as  to 
check  the  computations  of  the  leveler. 

Prob.  24.  Explain,  with  a diagram,  why  it  is  that  precision 
i in  levelling  is  promoted  by  setting  the  instrument  midway  be- 
tween the  turning  points. 

i, 

Art.  25.  Contours  and  Profiles. 

In  Art.  2 it  was  stated  that  the  dimensions  of  a field  are  the 
horizontal  projections  of  the  actual  boundary  lines  and  that 


LEVELING  AND  TRIANGULATION, 


76 

the  area  is  that  included  between  the  projections  of  the  bound- 
aries. It  is  evident  that  a map  made  under  these  conditions, 
while  giving  a clear  idea  of  the  shape  and  size  of  the  property, 
will  convey  no  information  as  to  the  character  of  the  surface, 
whether  high  and  uneven  or  fiat  and  low.  These  distinctions 
would  be  evident  if  the  elevations  of  very  many  points  in  the 
field  were  written  at  the  proper  places  on  the  map,  but  so 
many  figures  would  render  other  features  of  the  map  indistinct, 
and  hence  another  plan  of  indicating  the  elevations  has  been 
adopted.  If  the  surface  of  the  ground  were  cut  by  a series  of 
horizontal  planes  at  equal  distances  apart,  the  intersection  of 
each  plane  and  the  ground  would  be  an  irregular  line  connect- 
ing ail  points  having  the  elevation  of  that  plane.  These  inter- 
sections called  Contour  Lines,  are  plotted  on  the  map  and  show 
at  a glance  the  elevations  and  slopes  of  all  parts  of  the  field 
with  a precision  dependent  upon  the  nearness  of  the  planes  to 
each  other.  A clear  conception  of  the  utility  of  the  contour 
lines  as  the  means  of  judging  of  the  features  of  a surface  is 
formed  by  considering  the  surface  of  a lake  as  the  intersecting 
plane.  The  shore  line  is  the  contour  having  the  elevation  of 
the  surface  of  the  lake;  if  the  water  were  to  fall  a certain  dis- 
tance, the  horizontal  movement  of  the  shore  line  would  depend, 
not  only  upon  the  vertical  fall  of  the  surface  of  the  water,  but 
also  upon  the  declivity  of  the  ground,  being  small  where  the 
latter  is  steep  and  great  where  it  is  nearly  flat.  Hence  the 
slope  of  the  ground  is  judged  to  be  abrupt  where  the  map 
shows  the  contour  lines  near  together,  while  the  slope  is  slight 
when  they  are  far  apart. 

The  position  of  the  contour  lines  is  not  generally  located  in 
the  field,  but  elevations  are  taken  at  points  where  the  slope  of 
the  ground  changes,  or  often  at  stakes  set  at  regular  intervals 
by  the  transit  and  chain.  These  elevations  are  then  plotted  in 
pencil  on  the  map  and  the  positions  of  points  at  the  elevation 
of  any  contour  are  found  by  interpolating  between  two  plotted 
elevations  one  of  which  is  above  and  one  below  the  required 
point;  the  contour  lines  are  then  drawn  by  connecting  points  of 
equal  elevation  by  a curve;  the  elevation  of  the  contour  is 
marked  on  it  and  the  plotted  figures  erased.  Let  the  field 


CONTOURS  AND  PROFILES. 


77 


A BCD,  Fig.  39,  be  divided  into  squares  100  feet  on  a side  and 
[elevations  taken  at  all  the  corners  as  shown,  and  let  it  be  re- 
! quired  to  locate  the  even  ten-foot  contours.  Beginning  at  any, 
as  the  upper  right-hand  corner,  the  ground  along  the  upper 
line  is  seen  to  fall  from  elevation  133  to  122  in  100  feet,  hence 
the  130  foot  contour  is  T3T  of  the  length  of  the  square  from  the 
j corner,  and  the  120  foot  contour  is  seen  to  be  ^ 0 f the  distance 
j from  the  second  corner  toward  the  third.  In  like  manner  all 
the  lines  are  gone  over  and  the  contours  are  then  sketched  in. 


If  the  ground  is  very  uneven  many  complications  will  arise  in 
drawing  the  contours  from  the  plotted  elevations,  and  the  fol- 
lowing general  rules  will  be  useful  in  preventing  errors:  Con- 
tour lines  never  cross  each  other;  every  contour  on  one  side  of 
the  map  must  either  be  found  on  one  of  the  other  sides,  or  a 
second  time  on  the  first  one;  a contour  not  crossing  any  side  of 
the  map  is  one  continuous  line,  returning  into  itself ; a contour 
line  never  branches,  forming  a loop ; the  number  of  contours 
between  two  others  whose  elevations  are  alike  is  either  two, 
four,  or  some  other  even  number. 


78 


LEVELING  AND  TRI ANGULATION, 


The  intersection  of  the  surface  of  the  ground  by  a vertical 
surface  is  called  the  Profile  along  that  line.  The  profile  is 
made  by  taking  the  elevations  at  known  intervals  along  the  de- 
sired course  with  the  level;  these  intervals  are  plotted  to  any 
suitable  scale,  and  at  each  point  where  an  elevation  was  taken 
an  ordinate  is  laid  off  whose  length  is  the  elevation  at  that 
point.  The  utility  of  the  profile  is  increased  by  making  the 
vertical  larger  than  the  horizontal  scale,  as  by  so  doing  the  rel- 
ative differences  in  elevation  are  made  much  more  apparent. 
The  profile  is  very  important  in  determining  the  grade  and  the 
probable  expense  of  building  streets,  railroads,  sewers  and 
drains.  In  the  case  of  a street  profiles  of  the  middle  and  side 
lines  are  plotted  together,  using  ink  of  different  colors  if  neces- 
sary to  distinguish  the  three  lines,  and  the  suitable  position  for 
the  finished  grade  is  selected;  profiles  at  right  angles  to  the 
street  line,  or  cross-sections,  at  suitable  distances,  as  every  50 
feet,  are  plotted,  and  on  them  is  marked  the  position  of  the 
grade  line;  the  area  between  the  latter  and  the  surface  indi- 
cates the  amount  of  excavation  or  embankment  necessary. 

The  profile  of  any  line  on  a contour  map  can  be  drawn  with- 
out any  additional  field  work,  since  the  elevations  of  the  inter- 
sections of  the  line  and  the  contours  are  known  from  the  height 
of  the  contours  themselves.  Thus  the  profile  of  a line  through 
the  middle  of  the  upper  row  of  squares  in  Fig.  39  would  be 
made  by  first  drawing  the  line  in  pencil  across  the  map,  then 
the  elevation  at  the  right  end  is  130;  at  about  115  feet,  going 
toward  the  left,  the  elevation  is  120;  70  feet  further  110;  and  so 
on  across  the  map.  The  vertical  distances  on  a profile  are 
usually  plotted  on  a scale  from  5 to  20  times  as  large  as  the 
horizontal  scale. 

Prob.  25.  Draw  the  profiles  of  the  ground  along  the  lines 
AB  and  CD  in  Fig.  39,  making  the  vertical  scale  ten  times  the 
horizontal  scale.  Draw  also  the  profile  on  the  line  BC. 

Art.  26.  Adjustments  of  a Transit. 

The  adjustment  of  the  telescope  for  parallax,  described  in 
Art.  22,  must  be  made  every  time  it  is  used.  With  care  in 


ADJUSTMENTS  OF  A TRANSIT. 


'9 


handling  the  following  additional  adjustments  of  the  transit 
will  only  need  attention  at  rare  intervals,  hut  the  instrument 
should  be  frequently  tested  to  see  if  it  is  in  order. 

Plate  Bubbles. — The  plane  of  each  small  level  bubble  must 
be  parallel  to  the  horizontal  plate.  To  find  if  this  is  the  case, 
carefully  level  the  instrument,  turn  the  alidade  through  about 
180  degrees,  and  note  whether  the  bubble  is  still  in  the  middle 
of  the  scale.  If  not,  move  the  capstan  screws  at  the  end  of  the 
bubble  tube  until  one  half  the  apparent  error  is  corrected. 
Then  level  the  instrument  again  and  repeat  the  operation. 
The  other  plate  bubble  is  adjusted  in  the  same  way. 

Collimation.  — The  line  of  sight  must  be  perpendicular  to  the 
horizontal  axis  of  the  telescope.  To  find  if  this  is  the  case,  set 
up  the  transit  on  nearly  level  ground  and  sight  on  a well-de- 
fined distant  object,  reverse  the  telescope  and  place  a pin  about 
300  feet  from  the  instrument  in  the  opposite  direction ; revolve 
the  alidade,  sight  to  the  same  object,  reverse  the  telescope,  and 
note  if  the  line  of  sight  strikes  the  pin.  If  not,  set  another  pin 
in  the  line  of  sight  by  the  side  of  the  first,  measure  the  distance 
between  them  and  place  a third  pin  at  the  middle  of  that  dis- 
tance. Then  turn  the  capstan  screws  on  the  side  of  the  tele- 
scope until  the  vertical  cross-hair  has  moved  one  half  the  dis- 
tance from  the  second  to  the  third  pin.  Next  pull  up  all  the 
pins  and  repeat  the  operation  until  adjustment  is  secured. 

Horizontal  Axis.— The  horizontal  axis  of  the  transit  telescope 
must  be  parallel  to  the  horizontal  plate,  or  in  other  words  the 
standard  must  be  of  equal  height.  To  find  if  this  is  the  case, 
level  the  plate  bubbles,  elevate  the  telescope  as  high  as  prac- 
ticable and  sight  to  a sharply  defined  object,  depress  the  tele- 
scope and  mark  a point  on  the  ground  at  about  the  same  eleva- 
tion as  the  instrument;  then  revolve  the  transit  in  azimuth, 
sight  upon  the  same  object  and  mark  another  point  on  the 
ground.  If  these  points  do  not  coincide,  move  the  screws  at 
the  top  of  one  of  the  standards  until  the  vertical  hair  bisects 
the  distance  between  the  points.  Next  repeat  the  operation 
until  the  adjustment  is  perfect. 

Attached  Bubble. — The  attached  level  bubble  must  be  paral* 


80 


LEVELING  AND  TRIANG  ULATION. 


lei  to  the  line  of  sight  of  the  telescope.  To  ascertain  if  this  is 
the  case,  set  up  the  instrument  and  Tevel  the  telescope;  drive  a 
stake  A about  a foot  from  the  plumb-hob,  hold  a level  rod  upon 
it,  and  take  the  rod  reading  ax  by  sighting  through  the  large 
end  of  the  telescope,  or  by  measuring  to  the  end  of  the  middle 
of  the  axis  of  the  telescope.  Drive  another  stake  B about  400 
away  and  take  the  rod  reading  bx.  Next  set  the  instrument  as 
near  B as  possible,  take  the  rod  reading  &2  upon  it,  and  the 
rod  reading  a2  upon  A.  Now  if  ax  — bx  equals  a2  — b^ , the 
lines  of  sight  are  horizontal,  and  the  attached  bubble  is  in  ad- 


justment. If  not,  without  moving  the  level,  set  the  rod  on 
the  stake  A , clamp  the  target  so  that  the  rod  reads 

!($i  bi  — bx ), 

set  the  horizontal  cross-hair  on  the  target,  and  then  move  the 
bubble  into  the  middle  of  the  tube  by  the  screws  for  that  pur- 
pose at  the  end.  The  operation  is  then  to  be  repeated  until 
perfect  adjustment  is  secured.  This  is  called  the  peg  method 
of  adjustment. 

Vertical  Arc. — After  the  preceding  adjustments  are  made, 
the  vernier  of  the  vertical  arc  should  read  0°  00'  when  the  at- 
tached bubble  is  level.  If  this  is  not  the  case,  the  vernier  may 
be  moved  by  the  screws  at  its  ends  until  the  zero  points  coin- 
cide. This  adjustment  is  not  very  satisfactory,  and  instead  of 
making  it,  the  correction  may  be  noted  and  applied  to  each 
angle  when  it  is  read,  being  positive  for  angles  above  and 
negative  for  angles  below  the  horizontal  when  the  vernier  is 
too  far  toward  the  objective  end  of  the  telescope. 

Magnetic  Needle. — The  number  and  freedom  of  the  oscilla- 
tions of  the  needle  indicate  the  strength  of  its  magnetism.  If 
the  needle  becomes  sluggish  it  may  be  remagnetized  bypassing 
over  it,  toward  each  end,  the  pole  of  a magnet  by  which  that 


COMPARISON  OF  TRANSITS. 


81 


end  is  attracted,  returning  the  magnet  for  each  stroke  through 
a circle  of  about  one  foot  diameter.  The  straightness  of  the 
needle  is  tested  by  reading  the  angle  between  the  two  ends, 
first  with  the  needle  is  its  normal  position,  then  when  turned 
end  for  end;  the  difference  is  double  the  real  error  and  the 
needle  should  be  bent  by  that  amount.  After  the  needle  has 
been  straightened,  the  two  ends  will  be  180°  apart,  if  the  pin 
upon  which  it  rests  is  in  the  center  of  the  circle.  If  this  is  not 
the  case,  clamp  the  instrument  in  any  position  and  bend  the 
pin  till  fhe  ends  of  the  needle  are  opposite  corresponding 
points;  then  turn  the  instrument  through  90°  and  again  make 
the  correction. 

Prob.  26.  Give  the  reasons  for  each  of  the  above  adjust- 
ments, drawing  a figure  in  each  case. 


Art.  27.  Comparison  of  Transits. 

The  tests  of  the  telescope  and  its  attached  level,  described 
in  Art.  23,  may  be  applied  also  to  the  transit.  All  the  tests  of 
adjustments,  given  in  Art.  26,  should  likewise  be  made  upon 
a transit  which  the  engineer  is  about  to  purchase.  In  addition 
to  these  there  are  others  relating  to  the  graduated  circle  which 
will  here  be  explained.  It  is  often  incorrectly  assumed  that 
the  larger  and  heavier  the  instrument  the  more  accurate  work 
it  is  capable  of  doing.  There  is  some  truth  in  this  with  res- 
pect to  the  level,  but  very  little  as  respects  the  transit.  For 
ordinary  work  a transit  is  large  enough  if  it  has  a circle  four 
inches  in  diameter.  Such  a circle  can  be  made  to  read  to  half- 
minutes,  and  be  practically  as  easily  read  as  if  its  diameter 
were  six  inches.  Moreover,  the  extra  weight  of  the  larger  sizes 
does  not  materially  affect  the  stability  of  the  transit  as  that  is 
, mainly  governed  by  the  stiffness  of  the  tripod  and  head.  For 
the  purposes  of  the  land  surveyor,  a plain  transit, — that  is,  one 
without  attached  bubble  and  vertical  arc, — is  perhaps  sufficient. 
For  work  in  towns  and  cities  the  engineers’  transit,  which  has 
the  level  bubble  and  vertical  arc  and  also  two  verniers,  is  to  be 
preferred.  Unless  there  be  two  verniers  the  following  tests  of 
the  graduated  circle  cannot  be  made. 


82 


LEVELING  AND  TRIANGULATION. 


Angular  Distance  of  Verniers. — The  angular  distance  be- 
tween the  zeros  of  the  two  verniers  should  he  exactly  180  de- 
grees, hut  it  sometimes  varies  from  this  by  half  a minute,  owing 
to  lack  of  care  by  the  maker.  To  ascertain  its  amount  the  ob- 
server must  be  able  to  estimate  halves  or  quarters  of  a minute; 
this  is  not  difficult  if  the  two  lines  on  each  side  of  the  one  that 
apparently  coincides  are  also  regarded.  Vernier  A is  set  ex- 
actly at  0°  and  then  the  amount  which  vernier  B exceeds  or 
lacks  of  180°  is  read.  Next,  vernier  A is  set  exactly  at  20°  and 
the  amount  which  vernier  B exceeds  or  lacks  of  200°  is  read. 
The  process  is  continued  at  intervals  of  twenty  degrees  over 
the  entire  circle,  and  the  results  are  tabulated  in  the  second 
and  fourth  columns  of  the  table  below,  the  plus  and  minus 
signs  denoting  the  excess  and  deficiency  of  the  supplement  of 
the  angle  n as  read  on  vernier  B.  The  table  is  so  arranged 
that  the  values  of  n from  0°  to  180°  are  in  the  first  column, 
while  those  from  180°  to  860°  are  in  the  third  column,  and  the 
respective  discrepancies  for  the  two  parts  of  the  circle  are 
called  di  and  d2.  The  next  step  is  to  take  the  means  of  the 
corresponding  values  of  these  discrepancies,  observing  the 


n 

*x 

n 

d2 

+ d% 

d\  — d2 

2 

2 

0° 

- 45" 

180° 

+ 45" 

0".0 

- 45.0 

20 

- 15 

200 

+ 

+ 15  .0 

- 30.0 

10 

- 30 

220 

+ 30 

0 .0 

- 30.0 

60 

00 

240 

+ 45 

-4-22  .5 

- 22.5 

80 

- 15 

260 

+ 45 

+ 15  .0 

- 30.0 

100 

00 

280 

+ 30 

+ 15  .0 

- 15.0 

120 

+ 60 

300 

00 

+ 30  .0 

+ 30.0 

140 

+ 60 

320 

- 30 

+ 15  .0 

+ 45.0 

160 

+ 60 

340 

- 45 

+ 7 .5 

+ 52.5 

D = + 120.0. 

algebraic  signs,  and  place  them  in  the  fifth  column.  The  sum 
of  these  is  D — -f- 120".0,  and  the  angular  distance  of  the  ver- 
niers is  180  degrees  plus  one-ninth  of  D , or, 


Angular  distance  of  verniers  = 180°  + ID  = 180°  00'  13", 

which  shows  that  an  error  of  13"  exists.  A more  reliable  re- 
sult can  be  obtained  by  taking  readings  at  intervals  of  ten  do* 


COMPARISON  OF  TRANSITS. 


83 


grees  around  tlie  circle,  in  which  case  the  sum  D is  to  be  divided 
by  eighteen. 

Eccentricity. — If  the  center  of  the  alidade,  to  which  the 
verniers  are  attached,  does  not  coincide  with  the  center  of  the 
graduated  plate,  it  will  revolve  around  the  latter  in  a small 
circle.  When  the  vernier  is  on  a line  joining  these  centers 
there  is  no  error,  but  for  any  other  position  all  the  readings  are 
affected  by  a greater  or  less  error  of  eccentricity.  The  last 
column  in  the  above  table,  which  is  found  by  taking  the  means 
of  the  differences  of  the  two  sets  of  discrepancies,  shows 
roughly  the  errors  of  eccentricity.  From  it  there  appears  to 
be  no  error  when  vernier  A reads  about  105°  or  285°,  and  a 
maximum  error  at  about  160°  or  340°.  A closer  estimate  of 
these  quantities  can,  however,  be  made,  and  the  distance  be- 
tween the  two  centers  be  computed.  Let  each  of  the  quantities 
in  the  last  column  be  multiplied  by  the  sine  of  the  angle  in  the 
first  column  and  the  algebraic  sum  of  the  products  be  called  s. 
Let  each  quantity  be  also  multiplied  by  the  cosine  of  the  angle, 
and  the  algebraic  sum  of  the  products  be  called  t.  Using  only 
two  decimals  in  the  sines  and  cosines,  these  values  are  found 
to  be  s = — 20". 4 and  t = — 208'. 3.  Then  the  probable  angle 
n o at  which  no  error  of  eccentricity  exists  is  found  by 


whence  n0  = 95^°.  Also  the  probable  maximum  value  of  the 
error  of  eccentricity  is,  if  m be  the  number  of  readings  on  half 
the  circle, 

• E= ^ — = 46". 5. 

m sin  nQ 

Lastly,  the  radius  of  the  circle  in  which  the  center  of  the 
alidade  revolves  round  the  center  of  the  limb  is  to  be  found. 
Let  B be  the  radius  of  the  graduated  limb,  which  in  this  case 
is  2J  inches;  then  the  radius  of  eccentricity  is 

r = \BE  sin  1"  = 0.00028  inches, 
which  is  the  distance  between  the  two  centers.  Although  this 
is  a very  small  quantity,  it  yet  produces  sensible  errors  in  the 
readings. 

By  taking  several  sets  of  readings  in  the  manner  described, 


84 


LEVELING  AND  TRIANGULATION. 


a fair  idea  can  be  obtained  of  the  angular  distance  between  the 
verniers  and  of  the  effect  of  eccentricity  on  readings  in  different 
parts  of  the  circle.  The  theory  of  errors  of  eccentricity  is  not 
given  here,  as  it  belongs  properly  to  higher  surveying,  but  it 
has  been  thought  well  to  explain  the  method  of  procedure  in 
order  to  enable  the  owner  of  a transit  to  investigate  its  weak- 
nesses. It  fortunately  happens  that  in  precise  angle  measure- 
ments the  effect  of  these  sources  of  error  can  be  largely  elimi- 
nated by  the  method  of  repetitions  described  in  Art.  30. 

Prob.  27.  Test  two  transits  by  the  above  methods  and  write 
a report  giving  the  observations  and  computations  in  full,  and 
comparing  the  two  instruments. 


Art.  28.  Standard  Tapes. 

In  town  and  city  surveying  linear  measurements  of  a high 
degree  of  precision  are  often  necessary,  and  it  is  also  very  im- 
portant that  all  measures  should  be  referred  to  the  same  stand- 
ard. A steel  tape  duly  certified  by  the  Bureau  of  Weights  and 
Measures  at  Washington,  is  the  most  convenient  standard,  and 
it  should  not  be  used  for  any  purpose  except  for  the  comparison 
of  other  tapes.  The  standard  tape  is  certified  to  be  correct  a,t 
a given  temperature  when  under  a given  pull ; or  the  error  of 
its  length  is  stated  for  a given  temperature  and  pull.  The  co- 
efficient of  expansion,  or  the  relative  change  in  length  for  one 
degree  Fahrenheit,  should  also  be  stated  in  order  to  render 
comparisons  at  other  temperatures  possible.  For  example,  a 
certain  tape  400  feet  long  is  stated  to  be  a standard  at  56  de- 
grees Fahrenheit  when  under  a pull  of  16  pounds,  and  its  co- 
efficient of  expansion  is  given  as  0.00000703.  At  a temperature 
of  49  degrees  the  length  of  this  tape  will  be 

400  - 0.00000703  X 7 X 400  = 399.980  feet; 
at  a temperature  of  70  degrees  its  length  will  be 

400  + 0.00000703  X 14  X 400.  = 400.039  feet. 

To  compare  another  tape  with  the  standard  it  is  necessary  to 
know  its  coefficient  of  expansion  also.  In  order  to  determine 
this  the  tape  should  be  stretched  out  on  the  floor  of  a large 


STANDARD  TAPES. 


85 


room  whose  temperature  can  be  varied  or  be  kept  tolerably 
uniform.  With  a spring  balance  at  each  end  it  is  pulled  to  the 
proper  tension,  the  thermometer  noted,  and  a certain  length 
marked  on  two  tin  plates  temporarily  fastened  on  the  floor. 
The  temperature  is  then  raised  or  lowered,  and  the  operation 
again  repeated.  The  change  of  length  as  marked  on  the  tin 
plates  is  accurately  measured,  and  this  divided  by  the  total 
length  and  by  the  number  of  degrees  of  change  gives  the  co- 
efficient of  expansion.  For  example,  suppose  that  at  a temper- 
ature of  41  degrees  a length  of  60  feet  is  marked  off,  and  that 
this  is  done  again  at  a temperature  of  79  degrees,  the  pull  be- 
ing the  same  in  both  cases,  and  the  change  in  length  being 
0.016  feet.  Then  the  coefficient  of  expansion  is 

(0.016  -r-  60) -s-  (79  - 41)  = 0.00000701. 

Owing  to  the  delicacy  of  this  operation,  a single  result  is  not 
reliable,  and  hence  a number  of  observations  should  be  made 
under  different  conditions  and  the  mean  of  the  various  results 
be  taken  for  the  final  coefficient. 

The  operation  of  comparing  a tape  with  a standard  consists 
in  laying  off  the  same  distance  by  both  and  thus  determining 
the  temperature  at  which  the  former  is  correct.  The  pull  on 
the  tape  may  be  selected  to  agree  with  its  size,  but  the  pull  on 
the  standard  must  always  be  the  given  assigned  pull.  As  an 
example,  let  the  standard  be  exactly  400  feet  long  at  56  degrees 
Fahrenheit  when  under  16  pounds  pull,  and  its  coefficient  of 
expansion  be  0.00000703.  Let  the  tape  to  be  tested  be  300  feet 
long,  its  coefficient  of  expansion  being  0.00000690.  With  the 
standard  300  feet  is  laid  off  with  the  pull  of  16  pounds,  and  the 
temperature  is  noted  as  63  degrees.  With  the  tape  300  feet  is 
also  laid  off  under  a pull  of  18  pounds,  the  temperature  being 
noted  as  64  degrees.  The  second  distance  is  found  to  be  0.039 
feet  longer  than  the  first.  Now  let  t be  the  temperature  at 
which  the  tape  is  correct  under  18  pounds  pull,  then 

300[1  + 0.00000690(64°  - Q]  - 300[1  + 0.00000703(63°  - 56°)] 
= 0.039, 

from  which  t is  found  to  be  38  degrees.  The  tape  is  therefore 
a standard  at  38  degrees  Fahrenheit  when  under  18  pounds 


86 


LEVELING  AND  TRIANGULATION. 


pull,  and  a measurement  l made  by  it  at  any  other  temperature 
T will  have  the  true  value  l + 0. 00000690(  T — 38 °)l. 

If  the  tape  is  to  be  used  under  different  pulls  its  coefficient  of 
stretch,  or  relative  change  in  length  for  one  pound  pull,  should 
also  be  determined.  The  operation  for  doing  this  is  similar  to 
that  above  described  for  the  coefficient  of  expansion,  except 
that  the  temperature  should  be  constant  and  the  pull  be  varied. 
For  example,  let  a length  of  300  feet  be  marked  off  at  15 
pounds  pull  and  again  at  19  pounds  pull,  and  let  the  change  in 
length  be  0.026  feet.  Then  the  coefficient  of  stretch  is 
(0.026  -T-  300)  -7-  (19  — 15)  = 0.0000216.  Any  length  l made  un- 
{ ier  a pull  P,  other  than  the  standard  pull  of  18  pounds,  will 
then  have  the  true  value  l-\-  0. 000021 6(P  — 18)£,  provided  the 
standard  temperature  of  38  degrees  exists. 

Sometimes  the  tape  is  stretched  over  two  supports  A and  B, 

and  thus,  owing  to  the  sag,  the  measured  distance  is  too  long. 

_ Let  l be  the  distance  read  on  the 

?/  p 

U~  tape  under  a pull  P,  let  d be  the 

A B deflection  or  sag  at  the  middle, 

Fig.  41.  and  w the  weight  of  the  tape  per 

linear  foot.  The  curve  of  the  tape  is  closely  that  of  a parabola, 

8 d 2 

and  if  L be  the  horizontal  distance  L = l — - — , very  nearly. 

o t 

Also  taking  moments  at  the  middle  of  the  span  Pd  — %wl . 
Eliminating  d from  these  two  equations  the  adjusted  length  is 

1 / wl  \2 

found  L = l — 0(2^7  ^ For  example,  let  w = 0.0066  pounds 

per  foot,  P = 16  pounds,  and  l = 309.851  feet,  then  L — 
309.642  feet.  If  the  distance  AB  be  subdivided  into  n equal 
spaces  by  stakes  whose  tops  are  on  the  same  level  as  those 

at  A and  B,  then  L — l — - 1 — =,  f l.  For  instance,  if  n = 7, 
6\2 11PI 

then  for  the  above  data  L — 309.847  feet. 

To  recapitulate:  Let  t be  the  temperature  and  p the  pull  at 
which  a tape  is  a standard,  let  T be  the  temperature  and  P the 
pull  at  which  a measurement  l is  taken,  let  e be  the  coefficient 
of  expansion  and  s the  coefficient  of  stretch,  let  to  be  the 


BASE  LINES. 


87 


weight  of  the  tape  per  linear  foot,  and  if  sag  exists  let  n he  the 
number  of  equal  spaces  in  the  distance  l.  Then 

Correction  for  temperature  =r  -f-  e(T  — t)l; 

Correction  for  pull  = -}-  s(P  —p)l; 

Correction  for  sags  = — lm 

& 24:\nPr 

For  example,  let  t = 56  degrees,  p=  16  pounds,  e — 0.00000703, 
5 = 0.00001782,  w = 0.0066  pounds  per  foot;  let  a distance 
309.845  feet  be  measured  at  a temperature  of  49£  degrees  under 
a pull  of  20  pounds,  there  being  7 subdivisions  in  the  line. 
Then  the  correction  for  temperature  is  — 0.0142  feet,  that  for 
pull  -f-  0.0221  feet,  and  that  for  sag  — 0.0028  feet.  The  ad- 
justed measured  distance  is  hence  309.850  feet. 

Lastly,  if  the  measurement  is  made  upon  a slope  it  must  be 
reduced  to  the  horizontal  by  multiplying  it  by  the  cosine  of  the 
angle  of  slope.  It  is,  however,  generally  best  to  find  the 
difference  of  elevation  of  the  two  ends  of  the  line  by  leveling. 
1 f li  be  this  difference  and  L the  length  on  the  slope,  the  hori- 
zontal distance  is  |/A2  — A2.  For  instance,  if  the  length 
309.850  feet  has  2.813  feet  as  the  difference  of  level  of  the 
ends,  then  the  horizontal  distance  is  309.838  feet. 

Prob.  28.  A tape  is  a standard  at  41  degrees  Fahrenheit 
when  under  16  pounds  pull  and  no  sag,  its  coefficient  of  expan. 
sion  being  0.0000069  and  its  coefficient  of  stretch  0.000019. 
Find  the  pull  P so  that  no  corrections  will  be  necessary  when 
measurements  are  made  at  a temperature  of  38  degrees  and 
with  no  sags. 

Art.  29.  Base  Lines. 

A triangulation  necessarily  starts  from  a measured  base 
whose  length  must  be  known  with  precision  if  the  territory  to 
be  embraced  by  the  triangles  is  large.  A long  steel  tape,  duly 
standardized,  is  the  best  instrument  for  making  the  measure- 
ment. The  base  line  should  be  divided  into  divisions,  each 
shorter  than  the  length  of  the  tape,  and  stout  posts  be  set  at 
the  ends  of  the  base  and  at  the  points  of  division.  On  these 
posts  are  placed  metallic  plugs,  each  having  drawn  upon  it  a 


88 


LEVELING  AND  TRIANGULATION. 


fine  line  at  right  angles  to  the  direction  of  the  base.  The  ele- 
vations of  these  plugs  should  he  carefully  determined.  Each 
division  is  then  subdivided  into  equal  parts  by  light  stakes  set 
in  line  and  on  grade,  the  distance  between  the  stakes  being 
fifty  feet  or  less.  On  each  stake  two  small  nails  may  be  placed 
to  keep  the  tape  in  position. 

The  measurement  should  be  done  upon  a cloudy  day  with  lit- 
tle wind,  in  order  to  avoid  errors  due  to  change  in  temperature. 
The  tape  is  suspended  over  two  plugs  and  upon  the  interme 
mediate  stakes  and  pulled  at  both  ends  by  spring  balances  to 
the  desired  tension.  At  one  plug  a ten  foot  mark  on  the  tape 
is  made  to  coincide  with  the  fine  line  on  the  plug,  and  at  the 
other  end  a mark  is  made  on  the  tape  directly  over  the  fine  line 
on  that  plug.  The  odd  distance  can  then  be  measured  with  a 
separate  scale  to  the  nearest  thousandth  of  a foot.  Several 
measures  of  each  division  should  be  made  with  different  pulls, 
and  the  temperature  be  noted  at  each  reading. 


The  following  field  notes  of  a short  base  measured  by  stu 
dents  of  Lehigh  University  will  illustrate  the  method  of  opera 
tion.  There  were  three  divisions,  designated  as  I,  II,  and  IU, 


Division. 

No.  of  Sub- 
divisions. 

Difference 

in 

Elevation 
of  Ends. 

Tempera- 

ture. 

Pull. 

Observed 

Distance. 

Remarks. 

feet 

pounds 

feet 

Ill 

7 

2.813 

51° 

16 

309.865 

Base  EG. 

50* 

18 

309.857 

Oct.  3,  1888,  P.M. 

50* 

20 

309.842 

50 

16 

309.870 

50 

18 

309.857 

Cloudy,  with 

49* 

20 

309.845 

slight  wind. 

II 

7 

5.618 

48 

16 

332  736 

47* 

18 

332.727 

47* 

20 

332.712 

47 

16 

332.740 

47 

18 

332.726 

47 

20 

332.715 

I 

6 

7.924 

47 

18 

279.850 

47 

18 

279.843 

47 

20 

279.832 

48 

16 

279.848 

48* 

18 

279.840 

48 

20 

279.837 

the  first  having  six  and  the  others  seven  subdivisions.  The 
steel  tape  used  was  about  400  feet  long.  It  was  stated  by  the 


BASE  LINES. 


89 


makers  to  be  a standard  at  56  degrees  Fahrenheit  when  under 
a pull  of  16  pounds  and  having  no  sag.  By  a series  of  experi- 
ments its  coefficient  of  expansion  had  been  determined  to  be 
0.00000703,  its  coefficient  of  stretch  0.0000178$,  and  its  weight 
per  linear  foot  0.0066  pounds.  In  order  to  adjust  the  field  re- 
sults the  expressions  deduced  in  the  last  article  hence  are 

Correction  for  temperature  = — 0.00000703  (56  — T)l\ 

Correction  for  pull  = 0.00001782  (P  — 16)£; 

Correction  for  sag  = — 0.00001815  fT-0; 

from  which  the  corrections  are  computed.  For  example,  for 
division  III,  where  n = 7,  the  mean  of  the  observed  distances 


Temp. 

T. 

Pull 

Observed 

Corrections. 

Adjusted 

Distance. 

P. 

Distance. 

Temp. 

Pull. 

Sag. 

51° 
5oy2 
50  U 
50 
50 

4934 

lbs. 

16 

18 

20 

16 

18 

20 

feet 

309.865 

.857 

.842 

.870 

.857 

309.845 

feet 

- 0.0109 

- 0.0120 
- 0.0120 

- 0.0131 

- 0.0131 

- 0.0142 

feet 

0 

+ 0.0110 
+ 0.0221 
0 

+ 0.0110 
+ 0.0220 

feet 

- 0.0043 

- 0.0034 

- 0.0028 

- 0.0043 

- 0.0034 

- 0.0028 

feet 

309.850 

.853 

.849 

.853 

.8515 

309.850 

mean  = 309.856  mean  = 309.851 

n = 7 h = 2.813  feet  Final  horizontal  distance  = 309.838 


is  309.856  feet,  and  this  is  taken  as  the  value  of  l in  all  cases. 
The  corrections  being  found,  the  adjusted  inclined  distances 
are  obtained,  and  their  mean  309.851  is  the  value  of  the  in- 
clined length.  Lastly,  this  is  reduced  to  the  horizontal,  giving 
|/309.851‘2  — 2.813'2  — 309.838  feet  as  the  final  result. 

Proceeding  in  the  same  manner  with  divisions  II  and  I the 
corrections  are  found  and  the  sum  of  the  three  horizontal  dis- 
tances is  922.223  feet,  which  is  the  final  result  from  the  field 
work  above  given.  The  probable  uncertainty  of  this  result  is 
less  than  1 part  in  150,000,  which  shows  that  work  of  a high 
degree  of  precision  can  be  done  with  a steel  tape  whose  con- 
stants are  known. 

Prob.  29.  Compute  the  adjusted  inclined  lengths  and  the 
final  horizontal  lengths  of  divisions  II  and  I of  the  above  base 

line. 


90 


LEVELIXG  AXD  TRIAXGU LATIOX. 


Art.  30.  Triangulation. 

The  process  of  triangulation,  after  the  base  is  measured, 
consists  in  observing  the  angles  of  all  the  triangles.  The  data 
are  thus  at  band  for  computing  tbe  lengths  of  all  tbe  sides.  If 
tbe  azimuth  of  one  side  is  known,  or  has  been  obtained  by  tbe 
method  of  Art.  40,  tbe  azimuths  of  all  tbe  other  sides  are  easily 
found.  Lastly,  tbe  latitudes  and  longitudes  of  tbe  stations  of 
tbe  triangulation  are  computed  (Art.  3). 

In  triangulation  angle  measurements  are  required  to  have  a 
precision  greater  than  tbe  least  reading  of  tbe  vernier  will  give, 
and  tbe  method  of  repetitions  is  to  be  used.  To  illustrate  tbe 
principle  let  LOM  be  tbe  angle  to  be  measured.  Setting  tbe 
vernier  at  0°  00'  point  first  on  L , unclamp  tbe  alidade,  ana 
point  on  M.  Now,  without  reading  tbe  vernier,  unclamp  tbe 
limb,  point  on  A,  unclamp  tbe  alidade,  and  point  on  M.  Tbe 
vernier  has  thus  traveled  twice  ovei*  tbe  arc,  and  if  it  be  now 
read  tbe  value  of  tbe  angle  is  one  half  tbe  reading.  If,  how- 
ever, a third  repetition  is  made  before  reading,  tbe  value  of  tbe 
angle  is  one  third  of  tbe  final  reading.  Thus  tbe  effect  of 
repeating  an  angle  is  to  divide  tbe  error  of  tbe  vernier  reading 
by  tbe  number  of  repetitions.  More  than  four  repetitions  are, 
however,  not  usually  advisable,  since  tbe  effort  of  clamping  is 
to  introduce  a constant  tendency  to  error  in  one  direction. 

Tbe  process  of  repetition  in  any  important  case  should  be  so 
conducted  as  to  eliminate  tbe  effects  of  tbe  errors  of  non-adjust- 
ment, those  due  to  imperfections  of  tbe  graduated  limb,  and 
those  due  to  pointing  and  clamping.  Errors  due  to  lack  of 
level  of  tbe  limb  and  those  due  to  setting  tbe  instrument  or 
signals  in  tbe  wrong  position  cannot,  however,  be  eliminated, 
and  hence  great  care  should  be  taken  that  these  do  not  exist. 
Errors  due  to  collimation  and  to  tbe  horizontal  axis  of  tbe 
telescope  may  be  eliminated  by  taking  a number  of  repetitions 
with  tbe  telescope  in  tbe  direct  position  and  an  equal  number 
with  it  in  tbe  reverse  position.  Errors  due  to  angular  distance 
between  tbe  verniers  and  to  eccentricity  of  tbe  graduated  limb 
may  be  eliminated  by  reading  both  verniers  and  taking  their 
mean.  Errors  due  to  inaccurate  graduation  may  be  eliminated 


TRT  ANGULATION. 


91 


i 

by  taking  readings  on  different  parts  of  the  circle.  Errors  due 
to  pointing  and  clamping  may  be  largely  eliminated  by  taking 
one  half  of  the  repetitions  in  one  direction  and  the  other  half 
in  the  reverse  direction. 

The  following  form  of  field  notes  shows  four  sets  of  measure- 
! ments  of  an  angle  HOK,  each  set  having  three  repetitions. 
The  first  and  fourth  sets  are  taken  with  the  telescope  in  the 
direct  position,  the  second  and  third  with  it  reversed.  The 
first  and  second  sets  are  taken  by  pointing  first  at  II  and 
secondly  at  K,  the  third  and  fourth  are  taken  by  pointing  first 
at  K and  secondly  at  H.  At  each  reading  both  verniers  are 
read.  The  vernier  is  never  set  at  zero,  but  the  reading  before 
beginning  the  set  is  taken,  this  being  made  to  differ  by  about 
90  degrees  in  the  different  sets  so  as  to  distribute  the  readings 
ewer  the  entire  graduation.  After  completing  a repetition  both 
■verniers  are  again  read.  In  the  first  and  second  sets  the  mean 
final  reading  minus  the  mean  initial  reading  is  divided  by  3, 
the  lumber  of  repetitions,  to  give  the  angle  as  determined  by 
that  set,  In  the  third  and  fourth  sets  the  initial  reading  minus 
the  final  reading  is  divided  by  3.  If  very  accurate  work  is 
required  four  or  eight  additional  sets  may  be  taken  on  different 
parts  of  the  circle,  and  the  mean  of  all  will  be  the  probable 
value  of  the  angle. 


| Station 
| Observed. 

| No.  of  Reps. 

o 

Cl 

Eh 

Reading. 

Angle. 

o / / 

Remarks. 

0 i 

A 

B 

9 

Mean 

ft 

H 

20  04 

00 

30 

15 

Angle  at  station  O, 

3 

D 

62  25  10 

Sept.  30,  1895,  3 p m. 

K 

207  19 

30 

60 

45 

Brandis  Transit,  No.  716. 

H 

110  12 

30 

30 

30 

3 

R 

62  25  07 

John  Doe,  observer; 

K 

257  27 

60 

45 

52 

R.  Roe,  recorder. 

K 

350  02 

00 

15 

07 

Air  hazy,  no  wind. 

3 

R 

62  25  33 

H 

162  43 

15 

30 

22 

K 

80  56 

15 

00 

08 

80  + 360  = 440°. 

3 

D 

62  25  35 

H 

253  39 

00 

15 

22 

Mean  of  four  sets, 

i 

HOK  = 62p  25'  21". 

92 


LEVELING  AND  TRIANGULATION. 


In  repeating  angles  tlie  following  points  should  be  noted  : 
The  instrument  should  never  be  turned  on  its  vertical  axis  by 
taking  hold  of  the  telescope  or  of  any  part  of  the  alidade  ; the 
limb  should  never  be  clamped  when  the  verniers  are  read  ; the 
observer  should  not  walk  around  the  instrument  to  read  the 
verniers,  but  standing  where  the  light  is  favorable  he  should 
revolve  the  instrument  so  as  to  bring  vernier  A and  then  vernier 
B before  him  ; the  observer  should  not  allow  his  knowledge  of 
the  reading  of  vernier  A to  influence  him  in  taking  that  of  B ; 
care  must  be  taken  to  turn  the  clamps  slowly  and  not  too 
tightly.  If  these  precautions  be  taken  the  value  of  an  angle 


can  be  obtained  to  a high  degree  of  precision  with  a transit 
reading  only  to  minutes. 

The  stations  of  the  triangulation  should  be  points  which  are 
not  liable  to  be  lost,  such  as  holes  drilled  in  rocks  or  in  monu- 
ments firmly  planted  in  the  earth.  In  the  survey  of  a town, 
however,  some  points  may  be  used  upon  which  the  transit  can- 
not be  set,  as  for  instance  church  spires,  but  these  must  be  so 
selected  that  they  can  be  seen  from  many  other  stations.  Care 
should  be  taken  that  all  the  triangles  are  well  proportioned, 
and  in  general  this  will  be  secured  when  no  angle  is  less  than 
30  degrees  or  over  150  degrees. 

A triangulation  forms  the  framework  of  a map.  All  its  sta- 
tions being  accurately  located,  a traverse  may  start  at  any  one 
and  take  the  notes  necessary  for  a map  of  that  vciniity,  check- 


TRIANGU  L ATIOST. 


93 


ing  tlie  field  work,  perhaps,  by  ending  at  another  station.  Thus 
there  is  no  trouble  in  joining  different  surveys,  for  all  are  con- 
nected with  the  same  skeleton  framework.  In  plotting  the 
maps  a coordinate  system  of  lines  1000  feet  apart  is  first  drawn 
and  upon  it  the  triangulation  stations  are  located  ; from  these 
the  various  traverses  or  stadia  lines  are  laid  off  as  indicated  by 
the  field  notes.  The  precision  of  triangulation  work  will 
depend  upon  the  purpose  for  which  it  is  to  he  used  ; for  or- 
dinary town  or  topographical  surveys  it  will  perhaps  he  suf- 
ficient if  the  lengths  of  the  lines  and  the  coordinates  of  the 
stations  are  found  to  the  nearest  tenth  of  a foot. 

In  Fig.  42  is  represented  a small  triangulation  system  in 
which  EG  is  the  base  line  and  P a spire.  All  the  angles,  ex- 
cept those  at  P,  were  observed  by  the  method  of  repetitions, 
and  a part  of  the  final  results  of  the  computations  are  given  in 
the  table  below.  Here,  as  in  Chapters  I and  II,  the  azimuths 


Line. 

Azimuth. 

Distance. 

feet. 

Station- 

Latitude. 

feet. 

Longitude. 

feet. 

A Q 
AE 

186°  49'  38" 

404  57 

A 

2014.83 

3406.63 

25  36  07 

778.95 

E 

2717.30 

3743.23 

A P 

91  25  54 

593.55 

G 

2804  40 

4661.32 

E A 

205  36  07 

778.95 

H 

2458.20 

5379.37 

EG 

84  34  48 

922.22 

K 

2250.76 

5733.05 

E P 

160  18  15 

761.87 

M 

1290.02 

5266.68 

G P 

219  25  28 

1041.35 

N 

988.38 

4435.91 

GH 

115  44  28 

797.15 

Q 

1613.13 

3358.54 

HP 

251  37  29 

1453.48 

MP 

299  16  15 

1452.09 

are  counted  from  the  north  around  through  the  east,  south,  and 
west,  while  latitudes  are  positive  toward  the  north  and  longi- 
tudes positive  toward  the  east.  This  is  the  usual  method  in 
land  and  town  surveying.  It  should  be  said,  however,  that  in 
geodetic  work  and  in  extended  topographical  surveys  the 
azimuths  are  often  counted  from  the  south  around  through  the 
west,  north,  and  e^st,  while  latitudes  are  taken  as  positive 
toward  the  north  and  longitudes  as  positive  toward  the  west. 

Prob.  30.  Compute  the  latitude  and  longitude  of  P from 
the  above  data  by  several  different  methods. 


94 


TOPOGRAPHIC  SURVEYING, 


CHAPTER  IV. 

TOPOGRAPHIC  SURVEYING. 

Art.  31.  Large-Scale  Topography. 

The  scale  to  which  topographic  maps  are  drawn  depends 
upon  the  use  for  which  they  are  designed  ; if  it  is  desired  to 
show  a large  extent  of  territory  at  once,  the  scale  will  he  de- 
termined by  the  size  of  the  finished  map  which  will  be  most 
convenient  for  use  ; on  the  other  hand,  if  it  is  desired  to  show 
a smaller  territory  but  with  more  minuteness,  a l&rgdr  scale 
could  be  adapted  to  the  same  size  sheet  as  before.  The  scale 
of  the  map  influences  the  degree  of  accuracy  employed  in  the 
field  work  and  also  the  appearance  of  the  signs  used  in  repre- 
senting the  various  topographic  features. 

Under  the  term  ktrge  scale,  it  is  intended  to  include  maps 
plotted  to  a scale  larger  than  400  feet  to  an  inch.  Such  maps 
are  designed  to  show  the  contour  lines  with  from  2 feet  to  10 
feet  intervals,  the  former  distance  being  applicable  in  case  the 
country  is  flat,  and  the  latter  where  the  slopes  are  abrupt  or 
where  less  precision  is  required.  .All  rokds  and  -streets, 
whether  highways  or  on  private  property,  are  shown  and  also 
the  positions  of  the  property  lines.  Dwellings  and  other 
buildings  are  represented  in  their  true  shape  and  with  dimen- 
sions drawn  to  the  scale  of  the  map.  The  positions  of  isolated 
trees  are  located  by  measurement,  as  are  also  the  boundaries  of 
woods.  If  a stream  is  to  be  shown,  both  sides,  instead  of  the 
middle  line  alone,  are  plotted  unless  the  width  is  so  small  that 
one  stroke  of  the  pen  would*  cover  both  sides.  It  sometimes 
happens  that  objects  have  to  be  plotted  out  of  proportion  to 
the  rest  of  the  map  because,  mechanically,  it  is  impossible  to 
represent  them  on  the  proper  scale.  It  is  quite  impracticable 
to  plot,  or  for  the  eye  to  distinguish,  distances  on  the  map  of 
less  than  tTq  of  an  inch  ; if  the  scale  of  the  map  is  200  feet  to 
an  inch,  of  an  inch  represents  2 feet  and  hence  objects  of- 
less  size  than  that  are  indicated  by  one  line.  A specimen  of  a 
large-scale  topographic  map  is  shown  in  Fig.  43. 


LARGE-SCALE  TOPOGRAPHY. 


95 


96 


TOPOGRAPHIC  SURVEYING. 


The  conventional  signs  used  in  illustrating  topographic* 
characteristics,  whether  indicating  the  nature  of  the  ground  or 
of  the  crops  growing  upon  it,  are  designed  to  bear  some  degree 
of  resemblance  to  the  objects  they  are  to  represent ; the  motive 
in  the  use  of  the  signs,  however,  is  to  convey  information  con- 
cerning the  character  rather  than  the  actual  appearance  of  the 
objects,  and  hence  no  attempt  is  made  to  draw  the  signs  to  the 
scale  of  the  map,  other  than  to  make  them  of  such  size  and 
weight  as  will  harmonize  with  the  other  parts  of  the  drawing. 
It  is  of  the  first  importance  that  the  topographic  drafts- 
man be  entirely  familiar  with  the  exact  appearance  of  the 
signs  he  wishes  to  use  ; especially  is  this  true  if  the  drawing 
is  to  be  on  a large  scale  where  no  marks  are  made  at  random, 
but  each  one  is  to  perform  a definite  part  in  producing  the 
general  effect  of  the  whole.  Some  of  the  signs  in  most  fre- 
quent use  are  shown  in  the  sketches  given  in  Fig.  44. 

Care  must  be  taken  that  the  signs  are  so  made  as  to  avoid  a 
flat  appearance,  which  is  a common  fault  of  otherwise  well  ex- 
ecuted drawings.  It  is  a universal  custom  to  consider  the 
light  as  coming  from  the  direction  of  the  upper  left-hand  cor- 
ner, in  which  case  the  shadow  will  be  on  the  lower  and  right- 
hand  sides  of  the  figures,  and  accordingly  those  parts  are  made 
with  a somewhat  heavier  stroke.  In  making  the  signs  for 
grass  the  shade  is  very  slight,  except  in  swamps  where  the 
shadow  is  drawn  under  each  tuft,  but  in  case  of  the  forest  it 
is  of  great  importance  in  relieving  the  appearance  of  sameness 
which  the.  map  would  otherwise  have.  In  representing  water 
and  the  shore,  it  is  a common  fault  to  make  the  line  of  the 
latter  too  light,  the  distinction  between  this  line  and  the  first 
shade  line  of  the  water  should  be  very  marked. 

Scales  are  frequently  designated  as  ratios ; thus  a scale  of 
¥_i__  is  such  that  any  actual  line  in  the  field  is  25,000  times 
as  long  as  its  representation  on  the  map.  A scale  of  400  feet 
to  an  inch  is  the  same  as  4800  inches  to  an  inch,  or 
as  commonly  expressed. 

Prob.  31.  How  many  feet  are  represented  by  one  inch  on  a 
scale  of  TTroi)  • How  many  acres  are  represented  by  one  square 
inch  on  a scale  of  pi  u o ? 


LARGE-SCALE  TOPOGRAPHY, 


97 


I fjlifl  H $ * 

Horn 


O O G G O 
M,k  v ' U Cotton 


O O 


'*  o .o  ' ° G Q) 


O © 


jjt  . ^ ^ ^ y ^ 


u # # «^r  jfe 

3="  =•  — ]{ic0  '=r  — — w 

A-  A # # # ife  Ik 


Fig.  44. 


98 


TOPOGRAPHIC  SURVEYING. 


• Art.  32.  Small-Scale  Topography. 

In  surveys  covering  very  large  areas  the  details  are  made 
subordinate  to  the  general  features  of  the  country.  In  the 
previous  article  several  reasons  for  so  doing  were  stated,  and 
in  addition,  the  usefulness  of  the  maps  is  not  such  as  to  war- 
rant so  great  expenditure  as  would  be  involved  in  making  the 
maps  to  a large  scale.  The  saving  in  the  cost  is  due,  partly  to 
the  fact  that  less  labor  is  necessary  in  plotting  the  maps,  but 
more  especially  to  the  economy  of  time  possible  in  making  the 
survey,  since  objects  need  be  located  with  only  such  precision 
as  will  make  the  errors  on  the  map  unobservable.  The  smaller 
the  scale  the  less  frequent  will  be  the  revisions  necessary  to 
keep  the  maps  reliable  since  the  objects  subject  to  change  are, 
for  the  most  part,  omitted  on  the  small-scale  maps. 

The  topographic  maps  made  by  the  United  States  Coast  and 
Geodetic  Survey  and  by  the  United  States  Geological  Survey 
are  drawn  to  the  scale  of  1 to  62,500,  1 to  125,000,  or  1 to  250,- 
000,  with  corresponding  contour  intervals  of  5 to  50  feet,  10  to 
100  feet  and  200  to  250  feet.  These  scales  are  seen  to  be  ap- 
proximately one,  two,  or  four  miles  to  the  inch.  The  largest 
scales  are  used  where  the  country  is  most  densely  populated 
or  where  it  is  flattest.  Some  small-scale  maps  show  the 
streams,  the  state,  county,  and  town  divisions,  the  highways, 
railroads,  and  canals  ; but  private  ways  and  property  lines  are 
not  represented  ; features  of  public  importance  being  given, 
and  those  of  a temporary  nature  omitted. 

The  conventional  signs  used  on  the  small-scale  maps  are 
made  to  present  approximately  the  appearance  of  those  of 
larger  scales  when  seen  from  a distance  ; the  details  can  hardly 
be  distinguished  without  the  aid  of  a magnifying  glass. 
Buildings  are  represented  simply  by  black  rectangles  without 
much  regard  to  the  shape  or  size  of  the  houses  themselves. 
Isolated  trees,  small  orchards,  and  groves  are  not  shown,  but 
the  boundaries  of  forests  are  plotted  to  scale  and  the  interior 
is  filled  in  as  shown  in  ,Fig.  45,  with  signs  similar  to  those 
given  in  Fig.  44,  but  very  much  smaller.  The  highways  are 


SMALL-SCALE  TOPOGRAPHY, 


99 


Fig,  45, 


100 


TOPOGRAPHIC  SURVEYING. 


represented  by  parallel  lines  of  uniform  distance  apart,  with- 
out regard  to  the  actual  width  of  the  road.  The  scale  of  Fig. 
45  is  while  that  of  Fig.  58  is  being  taken 

from  the  maps  of  the  Coast  and  Geodetic  Survey. 

The  use  of  colors  is  not  as  frequent  as  formerly,  but  the 
appearance  of  any  map  is  improved  and  its  utility  increased 
by  the  contrast  thus  made,  if  the  land  be  covered  with  a light 
wash  of  burnt  sienna  with  the  contour  lines  of  a darker  shade 
of  the  same  color,  and  the  water  colored  blue  ; all  other  marks 
are  in  black. 

Prob.  32.  Draw  a profile  of  the  surface  as  cut  out  by  a 
vertical  plane  through  the  NE  and  aSTF  corners  of  Fig.  45. 

Art.  33.  Theory  of  the  Stadia. 

The  fundamental  principle  of  stadia  measurements  is  that 
of  similarity  of  triangles.  In  Fig.  46  let  T represent  a tube 
having  three  horizontal  hairs  and  let  vertical  graduated  rods 
be  held  in  the  positions  AB  and  A\B\.  The  eye  being  at  the 
end  E , the  distances  CE  and  CiE  of  the  rod  from  E are  directly 


Fig.  46. 


proportional  to  the  spaces  AB  and  AxBi  apparently  inter- 
cepted on  the  rods  by  the  cross-hairs.  This  simple  proportion 
is  modified  somewhat  in  practice  by  the  fact  that  a telescope 
replaces  the  plain  tube. 

In  Fig.  47,  the  cross-hairs  are  at  a and  b,  and  i is  the  dis- 
tance between  them.  Rays  of  light  supposed  to  pass  outward 
from  a and  b are,  by  refraction  of  the  object  glass,  made  to 
intersect  at  0 , at  a distance  from  the  lens  equal  to  the  focal 
length  of  the  telescope  ; these  rays  intersect  the  rod  at  A and 
By  the  points  upon  which  the  hairs  a and  b are  apparently 
projected  by  the  eye  at  E,  If  the  rod  is  moved  to  any  other 


THEORY  OF  THE  STADIA. 


101 


point  distant  d ' from  0 the  space  intercepted  on  the  rod  by  the 
cross-hairs  will  have  the  same  relation  to  AB  that  d does  to 
d,  because  of  the  similarity  of  triangles  as  in  Fig.  46.  The 
total  distance  from  the  instrument  to  the  rod  is  D = c d ; 

in  which  c is  the  distance  from  the  plumb-bob  to  the  object 
glass  and  F is  the  focal  length  of  the  telescope.  From  the 
figure  it  is  seen  that 

f 

d : AB  ::  / : i,  or  d = R~  ; 

J i 

hence  D = {c  +/)  + 

% 

From  this  equation  it  would  appear  that  the  determination  of 
D depends  upon  very  careful  measurements  of  f and  i,  but 


Fig.  47. 


such  measurements  are  impracticable  and  unnecessary  since 
the  value  of  can  be  determined  by  trial  when  c and  f are 

approximately  known.  The  distance  c is  found  by  measuring 
from  the  axis  of  the  telescope  to  the  middle  of  the  object 
glass  when  the  telescope  is  focused  for  a distance  of  about  300 
feet  or  a mean  of  all  the  distances  that  are  to  be  measured. 
When  the  telescope  is  focused  for  an  infinite  distance  f is  the 
space  between  the  object  glass  and  the  cross-hairs  ; this  can 
readily  be  measured  with  sufficient  accuracy  when  the  focus  is 

for  an  object  a mile  or  so  distant.  To  find  the  value  of  '4, 

measure  from  the  center  of  the  instrument  any  convenient 
distance,  as  (c  -| -/)  + 200  feet,  along  level  ground  and  hold 
the  rod  on  the  point  thus  found.  Sight  to  the  rod  and  count 
the  number  of  spaces  on  it  between  the  upper  and  lower  hairs, 

then  the  constant  number  A.  can  be  found  from  the  equation 


102 


TOPOGRAPHIC  SURVEYING. 


f 

D — (c+/)  + Rt-.  Thus  let  c — 5 inches,  f = 7 inches,  the 

measured  distance  to  the  rod  201  feet,  and  the  space  intercepted 
on  the  rod  2.02  feet ; then 

201  = (0.48  + 0.52)  + 2.02  =£, 


f _ 200 
i ~ 2.02 


= 99.01. 


This  would  he  a very  awkward  factor  to  use  and  hence  it  is 
desirable  to  either  change  the  value  of  i by  moving*  the  hori- 
zontal hairs,  or  to  substitute  another  rod  on  which  the  gradua- 
f 

tions  are  of  such  size  that  . multiplied  by  one  of  the  units 
will  equal  100. 


To  adjust  the  hairs  to  fit  the  rod,  measure,  on  nearly  level 
ground,  some  convenient  distance,  as(c+/)+200  feet  from 
the  plumb -bob,  and  sight  upon  the  rod  held  at  that  distance 
from  the  instrument  ; move  the  upper  hair,  by  means  of  tjie 
capstan  screw  for  the  purpose,  till  one  space  is  intercepted  on 
the  rod  between  the  upper  and  middle  hairs,  then  similarly 
apply  the  correction  to  the  lower  hair.  In  case  an  ordinary 
self-reading  level  rod  is  used  the  cross-hairs  would  intercept 
two  feet  on  it  when  the  distance  from  the  instrument  hi 
( c +/)  + 200  feet. 

If  the  cross-hairs  are  fixed,  the  rod  can  be  so  graduated  that 
the  number  of  spaces  intercepted  on  it  by  the  hairs  will 
always  be  the  number  of  hundred  feet  that  the  rod  is  from  a 
point  feet  in  front  of  the  instrument.  Sight  to  the 

plain  rod  held  at  a distance,  say,  (e  +/)  +.300  feet  from  the 
instrument  and  mark  where  the  upper  and  lower  hairs  inter- 
sect the  rod  ; this  space  divided,  in  this  case,  by  three  is  then 
the  unit  by  which  the  whole  rod  is  to  be  graduated.  After  the 
units  are  marked  on  the  rod  they  are  sub-divided  into  ten  or 
twenty  equal  parts  to  aid  the  eye  in  estimating  distances  other 
than  the  even  hundreds. 


When  the  rod  is  to  be  used  in  surveys  which  are  to  be 
plotted  to  a small  scale,  the  constant  (c  +/)  is  often  disre 
garded  and  the  rod  is  graduated  accordingly.  The  rod  is  held 
at  distance  from  the  plumb-bob  which  is  supposed  to  be  about 


THEORY  OF  THE  STADIA. 


103 


a mean  of  all  distances  to  be  measured,  and  so  graduated  that 
the  rod  reading  will  correctly  indicate  that  particular  distance. 
When  the  rod  is  held  nearer  the  instrument  the  indicated  dis- 
tance is  a little  too  small  while  distances  greater  than  the 
mean  are  slightly  too  large.  If  the  rod  is  graduated  for  500 
feet  the  maximum  error  for  distances  between  100  feet  and 
1000  feet  will  be  about  1 foot. 

If  the  rod  is  to  be  always  used  in  open  country  where  the 
whole  of  it  can  be  seen  the  following  method  of  graduation 
may  be  adopted.  Hold  the  rod  at  100  feet  from  the  instrument 
and  mark  the  space  intercepted  by  the  cross-hairs,  the  upper 
one  being  sighted  to  the  uppermost  mark  on  the  rod  or  the 
lower  one  to  the  lowest  mark  ; next  hold  the  rod  at  200  feet 
from  the  instrument,  direct  the  same  hair  as  before  to  the 
mark  at  the  end  of  the  rod  and  note  the  point  intersected  by 
the  other  hair.  The  graduations  for  the  entire  rod  are  made  in 
a similar  manner  by  marking  the  spaces  actually  intercepted 
at  each  successive  100  feet  distance  from  the  instrument,  one 
hair  always  being  on  the  beginning  of  the  graduations. 

When  the  line  of  sight  is  inclined  to  the  horizontal  it  is 
evident  that  the  distance  indicated  on  the  rod  is  not  the  re- 
quired horizontal  distance  from  the  instrument.  If  the  rod  is 
held  perpendicular  to  the  line  of  sight,  the  reading  will  indi- 
cate the  inclined  distance  from  the  instrument  to  it ; the  hori- 


zontal distance  can  then  be  found  if  the  angle  between  the 
line  of  sight  and  the  horizontal  is  known.  In  practice  it  is 
found  to  be  impracticable  to  hold  the  rod  at  right  angles  to 
the  line  of  sight ; it  is  hence  placed  vertical  and  an  expression 
is  found  by  which  the  horizontal  distance  is  computed  from 
the  rod  reading  and  the  measured  vertical  angle  v. 


104 


TOPOGRAPHIC  SURVEYING. 


In  Fig.  48,  AB  is  tlie  reading  on  tlie  vertical  rod  and  A'B' 
tliat  when  the  rod  is  perpendicular  to  the  line  of  sight.  Since 
the  angle  AOB  is  small,  no  appreciable  error  will  result  if 
A' AB  is  considered  as  90°;  then 

A'B'  — AB  cos  v. 

A'B'  indicates  the  distance  OP,  and  TP  = c +/+  OP. 

TS  = TP  cos  v = (c  -\-f  + AB  cos  v)  cos  v; 

D = {c  f)  cos  v + R cos2  v, 

when  B is  the  distance  indicated  by  the  rod  reading.  The 
term  (c  -f-/*)  cos  v may  always  be  taken  as  one  foot  without 
any  practical  error. 

The  difference  in  elevation  II  is  found  by  sighting  the 
middle  cross-hair  to  a point  on  the  rod  at  the  same  height  a 
above  the  ground  that  the  telescope  is,  and  observing  the  ver- 
tical angle  v.  Thus, 

PS  = TP  sin  v = (c  + /+  AB  cos  v)  sin 
or, 

H—  ( c /)  sin  ® + B sin  v cos  v. 

For  values  of  v less  than  4 degrees  the  terms  (c  -f-  f)  sin  v may 
be  neglected,  and  (c  +/)  may  generally  he  taken  as  one  foot. 

The  above  formulas  for  D and  II  would  be  tedious  to  apply 
in  each  case,  and  hence  Table  X is  given  to  facilitate  the  re- 
ductions. This  table  was  computed  by  Professor  Arthur 
Winslow  for  the  Geological  Survey  of  Pennsylvania.  As  an 
example  of  its  use  suppose  that  ( c + /)  for  the  instrument  is  1 
foot,  and  that  a certain  rod  reading  gives  680  feet  for  a verti- 
cal angle  of  5°  26'.  Then 

B = 0.99  + 6.8  X 99.10  = 674  9 feet; 

R = 0.09  + 6.8  X 9.43  = 64.2  feet; 
or,  D — 674  feet,  and  H — 64.1  feet,  if  the  value  of  ( c +/)  is 
not  taken  into  account. 

Prob.  33.  Let  (c  +/)  = 1.2  feet,  B = 450  feet,  v = 3°  32'. 
What  is  the  error  in  considering  that  D = B for  the  horizontal 
distance,  and  II  = B cos  v sin  v for  the  difference  in  elevation 
between  the  height  of  the  instrument  and  the  point  sighted  on 
the  rod? 


FIELD  WORK. 


105 


Art.  84.  Field  Work. 

Tlie  topographic  survey  of  a large  territory  is  preferably 
based  upon  a system  of  triangulation,  which  will  afford 
numerous,  checks  upon  the  stadia  traverses.  The  stations 
should  be  located,  not  only  to  secure  well-conditioned  triangles, 
but  also  so  that  they  may  be  of  the  greatest  use  to  the  topogra- 
phers. In  a flat  wooded  country  a triangulation  system  is 
carried  on  only  at  great  expense  of  erecting  towers,  and  in  such 
cases  it  is  sometimes  advisable  to  locate  the  permanent  refer- 
ence stations  by  means  of  carefully  conducted  traverses.  By 
whatever  method  they  are  established,  the  stations  should  be 
near  enough  together  to  furnish  means  of  verifying,  each  day, 
the  work  of  the  topographical  parties.  The  elevations  of  the 
stations  are  to  be  determined  and  other  bench  marks  estab- 
lished at  proper  intervals  by  precise  leveling,  in  order  that  the 
errors  arising  from  the  use  of  the  stadia  in  determining  heights 
may  be  confined  to  the  short  traverse  lines  between  the  princi- 
pal stations. 

The  transit  used  in  stadia  surveying  need  not  be  of  large 
size,  but  there  are  some  features  that  are  especially  essential 
in  instruments  for  this  purpose.  The  telescope  should  have  a 
perfectly  flat  field  of  view,  since  the  lines  of  sight  do  not  coin- 
cide with  the  optical  axis;  this  defect  furnishes  the  opponents 
to  the  use  of  the  stadia  with  their  strongest  argument.  The 
vertical  arc  should  be  of  superior  quality,  the  graduations 
being  upon  solid  silver,  and  there  should  be  means  of  adjust- 
ing the  vernier  so  that  the  reading  shall  be  zero  when  the  tele- 
scope is  level.  A telescope  having  fixed  stadia  hairs  gives 
the  best  results,  but  can,  of  course,  be  used  only  with  a 
specially  prepared  rod.  The  horizontal  circle  should  have  its 
graduations  numbered  continuously  from  O’  to  860°  in  the 
direction  that  azimuth  is  reckoned,  and  there  should  be  means 
of  setting  off  the  magnetic  declination  so  that  the  needle  may 
indicate  north  or  south  when  the  line  of  sight  is  in  the  true 
meridian. 

The  stadia  rod  may  be  of  the  target  variety  or  self  reading; 
somewhat  greater  accuracy  may  perhaps  be  attained  by  the 


106 


TOPOGRAPHIC  SURVEYING. 


use  of  targets,  but  the  self-reading  rods  are  the  ones  in  most 
common  use.  The  rods  are  of  pine  about  eleven  feet  long, 
half  an  inch  thick,  and  four  inches  wide;  they 


are  sometimes  stiffened  by  screwing  to  the  back 
a longitudinal  strip  one  and  a half  inches  or 
so  square,  and  the  ends  may  be  prevented  from 
splitting  by  a metal  band.  There  are  numer- 
ous designs  for  painting  the  divisions  so  that  they 
are  readily  distinguished  upon  sighting  through  the 
telescope.  In  Fig.  49,  is  shown  a sketch  of  a rod 
which  is  known  to  give  good  satisfaction;  the  100 
feet  marks  are  painted  red  and  may  or  may  not  be 
numbered;  the  other  points  are  black,  and  the 
background  white.  The  rod  should  be  somewhat 
wider  than  the  part  painted  black,  so  that  there  may 
always  be  a white  background  for  the  cross-hairs 
The'  graduations  on  the  rod  do  not  extend  to  the 
Fig.  49.  extremities,  but  stop  at  equal  distances  from  both 
ends,  usually  about  half  a foot,  it  is  then  immaterial  which 
end  of  the  rod  is  held  on  the  ground. 


A topographic  surveying  party  is  composed  of  a transit 
man  or  observer,  a recorder,  one  or  more  rodmen,  and  axmen, 
if  they  are  required.  In  open  country,  where  the  topography 
is  not  very  intricate,  one  observer  can  take  sights  as  fast  as 
two  or  even  three  rodmen  can  select  points,  and  the  amount  of 
territory  covered  in  a given  time  is  very  much  increased  by  the 
use  of  the  extra  rods  ; in  more  difficult  territory  the  dispatch 
with  which  the  work  is  done  depends  largely  upon  the  skill  of 
the  recorder  in  keeping  his  notes  and  sketches  in  proper  shape, 
and  but  one  rodman  is  necessary.  The  work  in  the  field  con- 
sists of  running  traverse-lines  between  triangulation  stations  ; 
at  each  of  the  transit  points  along  the  traverse  the  topography 
is  taken  within  a radius  of  500  feet  to  1000  feet  around  the 
entire  circle  in  azimuth.  The  traverses  are  so  run  that  when 
the  work  is  finished  the  entire  territory -within  the  limits  of 
the  survey  has  been  covered  by  these  circles.  Before  starting 
a traverse-line  between  two  stations  the  elevations  of  the  sta' 
tions,  the  distance  between  them,  and  the  azimuth  of  the  line 


FIELD  WORK. 


107 


joining  them  should  have  been  determined.  The  transit  is  set 
over  the  first  station,  with  the  vernier  at  the  azimuth  of  the 
line  to  the  next  triangulation  station,  and  the  telescope 
directed  to  some  point  on  that  line  ; the  instrument  is  then 
oriented,  and  the  line  of  sight  is  brought  into  the  meridian  by 
setting  the  vernier  at  zero.  The  needle  is  allowed  to  settle 
and  the  magnetic  declination  set  off,  if  there  is  an  arrange- 
ment for  so  doing  ; otherwise  the  reading  of  the  needle  should 
be  noted.  In  locating  the  contours  the  rod  is  held  at  every 
place  where  there  is  a decided  change  in  the  slope  of  the 
ground  ; in  surveying  a small  ravine  elevations  are  taken  along 
the  valley  and  along  the  top  of  the  slope  on  each  side.  In 
work  that  is  to  be  plotted  on  a large  scale  two  points  on  each 
building  are  located,  and  it  is  well  to  have  the  dimensions 
measured  with  a tape.  The  rodman  should  have  a knowledge 
of  what  it  is  desired  to  show  on  the  map,  so  that  he  need  not 
rely  upon  signals  from  the  observer  to  select  the  points  where 
observations  are  to  be  taken.  When  the  work  around  the  sta- 
tion has  been  completed,  the  rodman  selects  a suitable  place 
for  the  next  position  of  the  transit  and  drives  a stake  there. 
The  observer  reorients  the  transit  and  reads  the  distance  to  the 
next  stake  ; in  determining  the  azimuth  the  edge  instead  of  the 
flat  side  of  the  rod  is  turned  toward  the  instrument.  The 
transit  is  then  set  over  the  new  station  while  the  rodman  gives 
a backsight  on  the  last  one.  The  instrument  is  oriented  by 
directing  the  telescope  to  the  backsight,  with  the  vernier  read- 
ing the  back  azimuth  of  the  line  ; an  easy  way  to  find  what 
the  reading  should  be  is  to  add  180°  to  azimuths  less  than  that 
amount  and  to  subtract  180°  from  those  that  are  greater.  The 
rod  reading  and  the  vertical  angle  should  be  again  observed, 
and  the  mean  of  the  two  corrected,  horizontal  and  vertical  dis- 
tances is  taken  as  the  length  of  the  line  and  the  difference  in 
elevation ; the  reading  of  the  needle  may  be  used  to  detect  any 
large  errors  in  azimuth.  Below  is  given  the  manner  of  re- 
cording the  notes  on  the  left-hand  page  ; the  right-hand  page 
is  used  for  the  sketch,  which  should  show  all  objects  located, 
and  be  as  near  to  scale  as  possible.  If  the  sketch  is  well 
made,  the  points  where  the  rod  was  held  are  numbered,  and 


108 


TOPOGRAPHIC  SURVEYING. 


tlie  same  numbers  appear  in  the  column  of  stations  on  the  left 
page  without  any  other  explanation.  The  traverse  is  finished 


Survey  of H.  I.  at  M = 491 .7 

Instrument  at  M.  c + / = 1.00.  Sept.  24,  1898.  Elev.  of  M = 486.6 


Point. 

Azimuth 

Rod 

Reading. 

Vertical 

Angle. 

Hor. 

Distance. 

DifT. 

Elev. 

Elev. 

1 

84°  12' 

907 

- 4°  24' 

2 

117  05 

605 

7 18 

3 

314  42 

245 

- 0 47 

N 

246  10 

723 

3 12 

721.8 

+ 40.3 

526.9 

by  connecting  with  another  station  on  the  triangulation  system, 
which  station  should  be  occupied,  and  the  azimuth  of  the  last 
course  be  verified,  while  a check  is  also  obtained  on  the 
elevations. 

Prob.  34.?  Fill  out  the  blanks  in  the  above  field-notes  by  the 
help  of  Table  X. 


Art.  35.  Office  Work. 

The  stadia  readings  taken  between  stations  of  the  tvs 
verses  are  usually  reduced  in  the  field  by  the  assistance 
Table  X.  The  topographer  thus  has  the  elevations  of  the 
stations  and  is  able  to  check  his  work  whenever  it  is  possible 
to  connect  with  a station  of  known  elevation.  The  horizontal 
distances  to  minor  points  and  the  corresponding  differences  of 
level  are,  however,  often  left  to  be  filled  out  in  the  office. 
Graphical  methods  have  been  devised  for  making  these  reduc- 
tions, but  none  has  become  so  valuable  as  to  displace  the  gen- 
eral use  of  the  tables. 

The  work  of  making  the  map,  like  that  in  the  field,  is  based 
upon  the  triangulation  system,  the  stations  of  which  are  care- 
fully plotted  by  their  coordinates  as  described  in  Art.  10. 
The  traverse  lines  are  plotted  by  the  protractor,  as  by  this  way 
the  work  on  the  map  can  be  done  as  accurately  as  the  measure- 
ments were  made  in  the  field.  A suitable  protractor  is  one  of 
cardboard  12  inches  in  diameter  which  is  fastened  to  the  paper 


OFFICE  WORK. 


109 


by  weights,  with  the  0°  and  180°  marks  on  the  meridian  ; 
azimuths  are  transferred  to  any  part  of  the  map  by  means  of 
triangles  or  parallel  rulers.  If  the  work  is  carefully  done,  the 
traverse  lines  should  close  so  that  the  discrepancy  is  not  notice- 
able on  the  scale  to  which  it  is  plotted.  The  error  of  closure 
may,  with  proper  care,  be  kept  less  than  1 in  500,  and  much 
better  results  than  this  have  been  attained. 

After  the  traverse  lines  have  been  established  the  topography 
is  plotted  by  orienting  the  protractor  over  each  station  and 
pricking  off  all  the  azimuths  of  the  readings  around  it  ; the 
protractor  is  then  removed  and  the  corresponding  distances  are 
measured  on  the  proper  scale.  The  sketch  will  show  whether 
the  point  is  merely  to  locate  contours  or  is  on  some  object  to 
be  plotted  on  the  map;  in  the  latter  case  the  house  or  whatever 
the  object  is  should  be  drawn  as  soon  as  enough  points  on  it 
have  been  established,  and  all  superfluous  marks  erased  ; if 
only  the  elevation  is  needed,  that  is  written  lightly  in  pencil. 
The  contours  cannot  be  sketched  as  fast  as  the  elevations  are 
marked,  but  this  work  should  not  be  deferred  after  enough 
heights  have  been  plotted  to  do  it  intelligently. 

What  was  stated  in  Art.  16  about  the  lettering,  title,  merid- 
ian, and  border  applies  as  well  to  topographic  drawings  and 
need  not  be  repeated.  The  execution  of  the  topographic 
signs  is  of  utmost  importance  in  determining  the  appearance 
of  the  map.  While  experienced  draughtsmen  are  able  to  dis- 
pense with  such  help,  no  student  should  attempt  to  make  the 
conventional  signs  on  a map  without  having  before  him  a good 
copy.  The  tendency  always  is  to  make  the  signs  much  too 
large  and  without  definite  shape.  No  amount  of  practice  will 
suffice  where  a clear  knowledge  is  wanting  of  just  how  the 
figure  should  look. 

Prob.  35.  Draw  in  pencil  six  horizontal  lines  and  twelve 
vertical  lines  on  Fig.  43  at  equal  distances  apa'rt.  Then  make 
the  same  number  of  lines  on  drawing-paper  at  distances  apart 
three  fourths  as  great.  Copy  Fig.  43  on  the  reduced  scale. 
(As  an  exercise  in  contour  drawing  Fig.  56  may  be  also  copied, 
the  scale  being  enlarged  about  one-half.) 


110 


TOPOGRAPHIC  SURVEYING, 


Art.  36.  The  Plane  Table. 

Tlie  plane  table  is  a small  drawing-board  mounted  on  a tri- 
pod head  and  tripod  like  those  of  the  transit.  On  the  board  a 
sheet  of  paper  can  be  fastened  by  clamps.  On  the  paper  a 
heavy  ruler  may  be  placed  in  any  position.  This  ruler  is  fur- 
nished with  level  bubbles,  and  at  its  middle  has  a standard  on 
which  is  mounted  a telescope  provided  with  a vertical  arc  and 
an  attached  bubble.  The  board,  which  can  be  moved  in  azi- 
muth around  the  vertical  axis  of  the  tripod  head,  corresponds 
to  the  limb  of  the  transit,  while  the  ruler  with  its  attachments 
corresponds  to  the  alidade.  The  adj  ustments  of  the  plane  table 
are  in  principle  the  same  as  those  of  the  transit.  (Art.  26). 

Although  the  plane  table  is  an  ancient  surveying  instrument, 
it  is  but  little  used  except  for  topographical  work  based  upon 
a triangulation.  On  the  paper  are  plotted  the  stations  of  the 
tri angulation,  or  as  many  as  are  contained  in  the  area  covered 
by  the  paper  on  the  scale  used.  A common  scale  used  is  5 
so  that  on  a board  24  X 30  inches  in  size  an  area  of  nearly 
2 X 2|  miles  would  be  represented.  In  a thickly  settled 
country  a scale  of  goV 0 °ften  used. 

In  a topographical  survey  one  of  the  first  uses  of  the  plane 
table  is  to  locate  on  the  sheet  secondary  triangulation  points, 


such  as  spires,  tall  chimneys,  or  prominent  trees.  In  Fig.  50 
this  process  is  illustrated.  A and  B are  two  triangulation 
stations  which  are  plotted  on  the  sheet  at  a and  b,  and  it  is  re- 
quired to  locate  the  two  secondary  stations  C and  I).  The 


THE  PLANE  TABLE.  Ill 

table  is  first  set  at  A,  the  edge  of  the  alidade  ruler  placed  upon 
the  line  ab,  the  telescope  pointed  to  B,  and  the  table  clamped 
in  position.  With  the  edge  of  the  ruler  on  a the  telescope  is 
pointed  to  G and  to  D,  and  indefinite  lines  drawn  in  those  direc- 
tions. The  table  is  then  set  up  at  B,  the  edge  of  the  ruler 
placed  upon  the  line  ba,  the  telescope  pointed  to  A,  and  the 
table  clamped  in  position.  With  the  edge  of  the  ruler  on  b the 
telescope  is  pointed  to  G and  to  D,  and  indefinite  lines  drawn  in 
those  directions.  The  intersection  of  these  with  those  pre- 
viously drawn  at  A gives  the  points  c and  d,  which  are  the  loca- 
tions on  the  sheet  of  the  stations  C and  I). 

The  operation  of  placing  the  table  so  that  each  line  on  the 
sheet  is  parallel  to  the  corresponding  line  on  the  ground  is 
( ailed  orienting  the  table.  After  the  table  is  set  up  and  leveled 
it  must  always  be  oriented  ; one  method  of  doing  this  is  ex- 
plained above,  and  this  will  apply  whenever  the  table  is  placed 
over  a point  which  is  plotted  on  the  sheet  and  from  which 
other  plotted  points  can  be  seen.  The  alidade  is  often  pro- 
vided with  a magnetic  needle  which  will  give  an  approximate 
orientation,  the  edge  of  the  ruler  being  placed  on  a magnetic 
meridian  drawn  on  the  sheet,  and  the  table  moved  in  azimuth 
until  the  needle  points  to  AT  on  the  compass  limb. 

When  the  table  is  placed  at  a point  on  the  ground  not  plotted 
on  the  sheet,  it  is  to  be  oriented  in  general  by  the  three-point 
problem.  An  approximate  orientation  is  first  made  by  the  eye 
or  by  the  magnetic  needle.  Three  stations,  A,  B,  and  G,  being 
visible  and  plotted  on  the  sheet  at  a,  b,  and  c,  it  is  required  to 
locate  the  point  n corresponding  to  the  point  N over  which 
the  table  is  set.  Placing  the  alidade  ruler  on  a,  b,  and  c in  sue 
cession,  and  sighting  on  A,  B,  and  G,  lines  are  drawn  on  the 
sheet,  and  these  intersect,  if  the  table  is  not  truly  oriented,  so 
as  to  form  a small  triangle  of  error.  Now  the  angle  between 
the  lines  Aa  and  Bb  will  hot  be  sensibly  altered  by  the  slight 
movement  necessary  to  effect  orientation  ; hence  the  point  n 
must  lie  on  the  circumference  of  a circle  passing  through  a , b, 
and  the  point  of  intersection  of  these  two  lines.  Similarly,  the 
point  n must  be  on  a circumference  passing  through  a,  c,  and 
the  intersection  of  Aa  and  Go,  It  is  not  practicable  to  draw 


Y ji  / 1!  [:  | I 

112  TOPOGRAPHIC  SURVEYING. 

these  circles  on  the  sheet,  but  by  imagining  them  to  be  drawn  a 
close  estimate  of  the  point  where  they  intersect  can  be  made, 
and  n be  marked  on  the  sheet.  Now  place  the  edge  of  the 
ruler  on  this  point  n , and  also  on  a , move  the  table  until  A is 
seen  on  the  telescope  hair,  and  a closer  orientation  is  secured. 
Then  sighting  to  B and  G,  and  drawing  new  lines  Bb  and  Cc,  a 


smaller  triangle  of  error  results,  from  which  a better  position 
of  n is  found,  and  on  the  third  trial  the  triangle  of  error  should 
entirely  vanish,  thus  giving  both  a correct  orientation  and  the 
proper  location  of  n corresponding  to  N on  the  ground. 

It  should  be  remarked  that  if  the  table  is  set  up  within  the 
large  triangle  ABC ',  as  in  the  first  diagram  of  Fig.  51,  the 
point  n falls  within  the  triangle  of  error.  In  other  cases  it 
falls  outside  the  triangle  of  error.  If  N is  situated  on  the  cir- 
cumference of  a circle  passing  through  A,  B,  and  G,  the  prob 
lem  is  indeterminate,  and  another  station  D must  be  observed  in 
connection  with  two  of  the  others.  For  a fuller  discussion  of 
the  three-point  method  of  orientation  see  ‘ ‘ A Treatise  on  the 
Plane  Table,”  in  Appendix  No.  13  of  the  Report  of  the  U.  S. 
Coast  and  Geodetic  Survey  for  1880. 

After  the  plane  table  is  oriented  the  topography  for  several 
hundred  feet  around  the  station  is  put  in  with  the  help  of  the 
alidade  and  stadia  rods.  The  alidade  ruler  gives  the  direction 
of  any  object,  and  the  stadia  reading  its  distance,  so  that  it  may 
be  immediately  plotted  by  a scale  an(J  a pair  of  dividers.  For 
an  inclined  stadia  reading  the  vertical  angle  is  read,  and  the 
corresponding  horizontal  and  vertical  distances  at  once  taken 
from  a table,  the  latter  giving  the  elevation  of  the  observed 


THE  THREE-POINT  PROBLEM.1  IT  3 

point  above  tbe  table,  which  is  noted  on  the  sheet,  so  that  the 
contours  can  be  afterward  sketched.  In  fact,  all  tbe  operations 
are  similar  to  those  explained  in  Art.  33,  except  that  no  notes 
are  kept.  Traverses  may  be  run  along  roads,  or  into  localities 
where  no  triangulation  points  are  visible,  by  drawing  the  lines 
successively  on  the  sheet  and  moving  the  table  from  one  station 
to  another,  orienting  it  by  a back  sight.  Thus  the  entire  map 
is  finished  in  pencil  in  the  field.  The  theory  of  all  the  opera- 
tions is  simple,  but  the  practice  requires  some  skill  and  experi- 
ence, and  the  sheet  is  sometimes  liable  to  become  injured  by 
dust  or  rain.  Much  more  topographic  work  is  done  with  the 
transit  and  stadia  than  with  the  plane  table. 

Prob.  36.  Given  two  stations  A and  B which  are  plotted  on 
the  sheet  at  a and  b.  It  is  required  to  set  the  table  at  two 
other  points  B and  E,  and  to  locate  d and  e on  the  sheet  by 
sighting  on  A,  B,  B,  and  E. 


Art.  37.  The  Three-Point  Problem. 

The  problem  of  determining  the  location  of  a point  by  means 
of  reading  the  angles  between  three  known  stations  is  of  fre- 
quent occurence  in  secondary  triangulation  work.  Thus  in 
Fig.  52,  let  Pi  P2  and  P3  be  three  stations  which  have  been 


fully  located  so  that  the  lengths  and  azimuths  of  the  lines  join- 
ing them  are  known,  as  also  their  coordinates.  At  a point  P 
the  transit  is  set,  and  the  angles  PjPP2  and  P2PP3  are  meas- 


114  TOPOGRAPHIC  SURVEYING. 

ured.  It  is  required  to  find  tlie  lengths  and  azimuths  of  PPi, 
P P 2)  PP3,  as  also  the  coordinates  of  P. 


Let  a and  b be  the  lengths  of  the  two  lines  PXP3  and  P2P3 
let  a and  be  the  angles  measured  at  P opposite  to  a and  b 
respectively,  let  dx,  d2,  d3  be  the  required  distances  from  P to 
Pi,  P2,  P3  respectively,  and  let  y be  the  known  angle  between 
PiP2  and  P2P3.  Let  x and  y represent  the  unknown  angles 
at  Px  and  P3.  Then 


x + y = m°  -(a  + /3  + y); 


a sin  x _ b sin  y 
sin  a sin  * 


Now  the  sum  x -f-  y is  known,  let  it  be  called  28;  also  let  the 
unknown  difference  x — y be  called  2 T.  Then  by  solving  the 
two  equations  the  following  method  results  : First,  compute  V 


from 


tan  V = 


a sin  ft 
b sin  a 


Secondly,  compute  Pfrom 

tan  T = cot  ( F+  45°)  tan  8, 


and  then  the  angles  x and  y are  found  from  : 
x = S+T ; y — S — T. 
The  required  distances  then  are  computed  by 


dx 


sin  (a+x)  sin  as  _ sin  y sin(/S  + y) 

■ . , U2  — Cl  — — 0 — ; ^3  — 0 : j, . 

sin  a sin  a sin  p sin  p 


The  azimuths  Zx,  Z2,  Z3  of  dx,  d 2,  d3  at  P are  now  found  by 
simple  addition  and  subtraction.  Let  Lx  and  Mx  be  the  lati- 
tude and  longitude  of  Pi,  and  L3  and  M3  those  of  P3.  Then 


L — Lx  — f-  dx  cos  Zx  — L3  -}-  d3  cos  Z3 ; 
M — jyfi  — dx  sin  Zx  — J\£3  — |-  d3  sin  Z3 ; 


which  afford  two  ways  of  computing  the  latitude  and  longitude 
of  P 

As  a numerical  example  let  the  following  be  the  data  as  de- 
termined* by  triangulation  for  three  stations,  the  azimuths  be- 
ing counted  from  the  north  as  in  Art.  30. 


Line.  Azimuth.  Distance. 

AB  147°  06'  49"  9011.0  ft. 

BC  254  56  58  5794.5 

CA  4 25  52  9098.9 


Station.  Latitude.  Longitude. 

A 34104.2  27418.4 

B 26537.2  32311.2 

O 25032.5  267155. 


HYDROGRAPHIC  SURVEYING.  115 

At  a point  P,  within  the  triangle  ABC,  there  are  measured  the 
angles  APB  = 127°  47'  33",  BPC  = 87°  38'  18",  CPA  = 144° 
34'  09".  It  is  required  to  compute  the  lengths  and  azimuths 
of  PA,  PB,  PC,  and  also  the  coordinates  of  P. 

Let  station  A correspond  to  Px  and  station  B to  P3 ; then  by 
comparison  with  Fig.  52  the  data  are  a = 144°  34'  09",  fi  = 87° 
38'  18  ',  ^ = 254°  56'  58"  - 184°  25'  52"  = 70°  31'  06",  a = 9098. 9 
feet,  b ~ 5794.5  feet.  Then  S = 28°  38'  14"  = i{x  + y).  First 
tan  V is  found  to  be  -f-  2.70529,  whence  V = 69°  43'  13".  Next 
[an  T is  found  to  be  — 0.25129,  whence  T = — 14°  06'  41". 
Then  x = 14°  31'  33"  and  y = 42°  44'  55".  The  distances  d\, 
d2,  d$  are  now  found  by  the  above  formulas,  the  two  values 
for  d2  being  3936.4  and  3936.6  feet,  which  agree  sufficiently 
well.  The  azimuth  of  PB  is  254°  56'  58"  + 42°  44'  55"  - 180°, 
and  those  for  PB  and  PC  are  also  easily  found.  Thus  there 
results: 

For  PA,  Azimuth  = 349°  54'  19",  Distance  — 5600.8  feet. 

For  PC,  Azimuth  = 205  20  10,  Distance  = 3936.5  feet. 

For  PB,  Azimuth  = 117  41  53,  Distance  = 4417.4  feet. 

Lastly,  the  values  of  dx  and  d3  are  multiplied  by  the  cosine 

and  by  the  sine  of  their  azimuths,  giving  the  differences  of 
latitude  and  longitude,  which  being  added  to  or  subtracted 
from  the  latitudes  and  longitudes  of  A and  B,  furnish  the  lati- 
tude and  longitude  of  P in  two  ways.  The  latitude  of  Pis 
28590.3  feet  and  its  longitude  is  28400.0  feet. 

Prob.  37.  Compute  the  azimuths  and  lengths  of  PA,  PB, 
PC,  from  the  above  data,  taking  A as  Pi,  B as  P2,  and 
C as  P3.  ' 

Art.  38.  Hydrographic  Surveying. 

When  a topographic  survey  embraces  rivers,  harbors,  or  a 
part  of  the  coast,  the  shore-lines  are  located  and  plotted  by  the 
methods  above  described.  It  is  also  generally  necessary  to  in- 
dicate on  the  map  the  depths  of  water  at  various  points,  the 
position  of  shoals,  rocks,  and  other  sub-surface  features,  and 
also  sometimes  to  determine  the  direction  and  velocity  of  cur- 
rents; this  part  of  the  work  constitutes  hydrographic  surveying. 


vo  / n c»  | 

116  TOPOGRAPHIC  SURVEYING. 


Soundings  in  shallow  water  are  made  by  means  of  rods  gradu- 
ated to  feet  and  tenths.  When  the  current  is  not  rapid,  a boat 
may  be  rowed  at  a uniform  speed  in  a straight  line,  which  is 
determined  by  signals  set  in  range  on  shore,  and  soundings  be 
taken  at  uniform  intervals  of  time.  The  position  of  the  boat 
both  at  the  start  and  finish  is  located  by  intersections  from 
other  signals  on  shore  or  by  means  of  observations  with  tran- 
sits When  this  line  is  plotted  on  the  map,  it  is  divided  into 
the  same  number  of  spaces  as  there  were  time  intervals,  and  at 
each  point  of  division  the  corresponding  sounding  is  plotted. 
If  the  number  of  soundings  is  sufficient,  contour  curves  for  dif- 
ferent depths  below  the  water-level  may  be  drawn,  and  thus  a 
clear  picture  is  presented  of  the  bottom  surface  of  the  river  or 
harbor. 

In  deep  water  where  a rod  cannot  be  used  depths  are  obtained 
with  a plummet  attached  to  a line,  the  position  of  each  sound- 
ing being  located  by  angles  taken  either  on  the  boat  between 
signals  on  the  land,  or  by  observers  on  shore.  In  the  former 
case  the  sextant  is  generally  used,  two  angles  being  measured 
between  three  known  stations.  This  is  a case  of  the  threei- 
point  problem  (Art.  37).  In  plotting  the  position  from  the  two 
observed  angles  computations  are  rarely  necessary,  but  throe 
lines  may  be  drawn  on  tracing- cloth,  intersecting  at  a point  and 
making  with  each  other  the  given  angles  ; then  placing  the 
tracing  on  the  map  so  that  the  three  lines  pass  through  the 
given  stations  the  point  will  fall  in  the  proper  position  and  may 
be  pricked  through  upon  the  map. 

In  all  cases  of  sounding  a water-gauge  should  be  erected  near 
the  shore  for  the  purpose  of  observing  the  variations  in  the 
water-level,  and  thus  referring  the  soundings  to  the  same  plane, 
either  of  high  or  of  low  water.  In  tidal  streams  or  harbors  read- 
ings of  such  a gauge  are  necessary  at  quarter-hour  intervals. 

The  sextant  is  a most  useful  instrument  in  all  work  done  in 
the  boat,  where  indeed  measurement  of  angles  with  a transit 
would  be  almost  impracticable.  The  principle  of  its  use  is  that 
an  object  may  be  seen  both  by  direct  vision  and  by  reflection 
from  a mirror.  For  instance,  in  the  first  diagram  of  Fig.  53  let 
H and  I be  two  parallel  mirrors  called  the  horizon  glass  and 


HYDROGRAPHIC  SURVEYING, 


11? 


Fig.  53. 


XiA  i 


1L8 


TOPOGRAPHIC  SURVEYING. 


the  index  glass,  the  upper  part  of  II  having  an  opening  in  it. 
Then  the  eye  at  E can  see  a distant  object  S,  both  by  direct 
vision  in  the  line  SEE,  and  by  the  reflected  ray  which  follows 
the  path  SIEE;  in  this  position  the  two  images  coincide  and 
the  index  arm  IA  indicates  zero  on  the  graduated  limb.  In  the 
second  diagram  the  index  arm  is  moved  to  the  position  ID  in 
order  to  measure  the  angle  SET,  between  two  signals  S and  T ; 
in  this  position  T is  seen  by  direct  vision  and  S by  reflection. 
As  the  angles  of  incidence  and  of  reflection  are  equal  on  each 
mirror,  the  angle  AID  is  one  half  the  angle  SET.  The  arc  is 


hence  graduated  so  that  half  a degree  on  it  represents  a whole 
degree  of  the  measured  angle  ; thus  the  reading  at  D gives  at 
once  the  required  angle  SET.  • 

In  measuring  a horizontal  angle  the  plane  of  the  sextant 
should  be  kept  as  nearly  horizontal  as  possible.  Care  should 
be  taken  that  the  reading  of  the  vernier  is  zero  when  an  object 
is  viewed  both  by  direct  and  reflected  vision,  as  in  the  first  dia- 
gram of  Fig.  54 ; if  this  is  not  the  case,  the  index  error  should 
be  noted  and  be  applied  as  a correction  to  the  final  reading. 

The  direction  of  currents  may  be  noted  by  observing  with  the 
sextant  the  direction  taken  by  a float  thrown  from  a boat,  and 
tile  velocity  of  the  current  may  be  found  by  noting  the  time 
required  for  the  float  to  pass  over  a certain  distance.  The  de- 
termination of  velocities  at  points  below  the  surface,  and  the 
gauging  of  streams  to  ascertain  their  discharge  and  mean  veloc- 


Fig.  54. 


MINE  SURVEYING. 


119 


ity,  is  properly  a brancli  of  hydraulics  rather  than  of  surveying. 
Concerning  these  see  Merriman’s  Treatise  on  Hydraulics  (New 
York,  1895),  Chapter  IX. 

Fig.  53  shows  a part  of  a hydrographic  map  of  the  Delaware 
River  on  a scale  of  sofoo>  reproduced  from  the  chart  of  the  U. 
S.  Coast  and  Geodetic  Survey.  The  numbers  in  the  central 
part  of  the  river  show  the  depths  in  fathoms  at  mean  low- water 
spring  tides,  those  on  the  shaded  surface  show  depths  in  feet. 
The  various  lights  and  buoys  are  represented  in  proper  posi- 
tion. The  topography  of  the  shores  is  a fine  example  of  small 
scale  work,  although  the  copy  does  not  fully  represent  the 
beauty  of  the  original  copper-plate  chart. 

Prob.  38.  Prove  that  in  Fig.  54  the  angle  AID,  moved 
over  by  the  index  arm,  is  one  half  the  observed  angle  SET. 


Art.  39.  Mine  Surveying. 

Mine  surveying  is  little  more  than  ordinary  surveying,  ren- 
dered difficult  by  darkness  and  mud.  The  main  object  is  to 
take  measurements  which  will  furnish  accurate  maps  of  the 
underground  workings,  so  that  the  position  of  every  point  may 
be  known  relatively  to  points  on  the  surface.  These  maps  are 
necessary,  both  for  the  advantageous  development  of  the  mine 
in  driving  tunnels,  slopes,  and  gangways,  and  for  the  safety  of 
the  miners.  The  maps  of  the  anthracite  coal  regions  of  Penn- 
sylvania are  required  by  law  to  be  drawn  on  a scale  of  100  feet 
to  1 inch,  and  to  be  kept  up  as  the  work  progresses. 

Mine  maps  show  the  main  features  of  the  surface  of  the 
ground,  such  as  streets  and  houses,  with  all  the  breakers, 
slopes,  manway  and  air-shaft  openings.  The  underground 
workings  are  shown  in  horizontal  projection  and  proper  posi- 
tion on  the  same  sheet,  different-colored  inks  being  sometimes 
used  to  distinguish  the  different  veins.  Elevations  of  many 
points  of  the  underground  workings  are  given  in  figures,  so 
that  the  difference  of  level  between  them  and  the  surface  is 
at  once  known,  as  well  as  the  grades  of  the  gangways  and 
other  passages.  Sometimes  the  surface  contours  are  also  shown, 


120 


TOPOGRAPHIC  SURVEYING. 


and  by  the  help  of  these,  and  the  elevations  of  the  underground 
points,  profiles  and  cross-sections  may  be  drawn  on  different 
vertical  planes. 

The  general  methods  of  mine  surveying  are  the  same  as 
those  of  land  and  topographical  surveying.  The  most  approved 
plan  is  to  have  on  the  surface  triangulation  stations  referred  to 
a system  of  coordinates  (Art.  30).  At  some  mines,  however, 
coordinate  lines  are  actually  staked  out  on  the  surface.  Start- 
ing at  any  station,  a traverse  may  be  run  down  a slope  and 
through  a gangway,  coming  out  perhaps  at  another  slope  or 
manway,  and  checking  on  another  triangulation  station.  This 
traverse  is  run  by  the  transit  and  a long  steel  tape,  two  con- 
secutive stations  of  the  traverse  being  generally  nearer  to- 
gether than  the  length  of  the  tape.  Offsets  are  taken  to  the 
sides  of  the  slopes  and  gangways,  and  short  lines  are  run  up 
the  breasts  and  openings.  Thus  all  the  data  are  obtained  for 
computing  the  traverse  and  constructing  the  map.  Elevations 
are  determined  by  taking  vertical  angles,  although  when  con- 
venient  the  level  and  rod  is  sometimes  used. 

The  stations  of  the  underground  traverse  are  placed  in  the 
roof  on  wooden  plugs  driven  into  holes  drilled  for  that  pur- 
pose. On  these  are  hung  the  plummet  lamps  to  which  back- 
sights and  foresights  are  taken.  To  set  up  the  transit  at  a 
station  a point  on  the  floor  directly  beneath  the  one  in  the  roof 
is  determined  by  the  plumb-bob.  A transit  for  mine  surveys 
should  have  a shifting  plate  and  adjustable  tripod  legs,  while 
a universal  joint  is  also  often  a great  convenience.  To  illumine 
the  cross- wires  the  transitman  holds  his  copper  lamp  at  arm’s- 
length  so  that  the  light  may  shine  into  the  objective  end  of  the 
telescope  ; the  same  lamp  enables  him  to  read  the  vernier  and 
the  magnetic  needle.  The  readings  of  the  magnetic  needle, 
which  serve  as  checks  on  the  horizontal  angles,  must  be  taken 
both  backward  and  forward  at  each  station,  as  marked  local 
attractions  occur  in  mines.  Much  time  is  often  wasted  in 
reading  the  needle  ; instead  it  would  be  better  to  check  the  azi- 
muth by  taking  another  angle.  The  linear  measurements  are 
made  when  the  tape  is  tightly  stretched  by  two  men,  offsets 


MINE  SURVEYING. 


121 


Fig.  55. 


122 


TOPOGRAPHIC  SURVEYING. 


being  taken  to  the  corners  of  pillars  and  the  sides  of  the  gang- 
ways. A mine  survey  corps  usually  consists  of  four  or  five 
men,  a transitman,  two  ckainmen,  and  one  or  two  men  for  off- 
sets and  lights. 

The  form  of  field-notes  may  be  the  same  as  that  given  in  Art 
15,  but  instead  of  measuring  the  interior  angles  it  is  best  to 
carry  on  the  azimuths  as  explained  in  Art.  19.  Some  prefer 
to  reverse  the  telescopes  and  measure  the  deflection  angle  to  the 
right  or  left,  but  this  is  inferior  in  accuracy  and  convenience 
to  the  method  of  azimuths.  The  form  of  notes  is  subject  to  so 
great  variations  in  different  localities,  that  it  seems  scarcely 
wise  to  attempt  to  give  one  of  them  here. 

The  computation  of  the  coordinates  of  the  stations  of  the 
traverse  is  next  made.  Lines  being  drawn  on  the  paper  500 
feet  apart  both  vertically  and  horizontally,  the  stations  are 
plotted  in  their  proper  positions.  The  offsets  are  then  laid  off 
and  the  sides  of  the  slopes,  gangways,  air-passages,  and  breasts 
are  drawn.  The  underground  traverse-lines  are  usually  plotted 
in  red,  and  each  station  designated  by  its  letter  or  number. 
The  elevations  are  noted  in  figures  at  such  stations  where 
they  may  be  likely  to  be  needed.  If  surface  features  are 
to  be  also  given,  they  are  plotted  from  the  notes  of  an  outside 
survey. 

Fig.  55  shows  a part  of  a map  of  an  anthracite  coal  mine,  re- 
duced from  the  original  scale  of  100  feet  to  1 inch  to  about 
half  that  scale.  It  shows  the  buildings  around  a slope  entrance, 
and  the  slope  with  a few  gangways  and  breasts.  The  fine 
broken  lines  are  the  traverses  of  the  survey  and  each  station 
has  its  number  ; a traverse  is  seen  to  start  at  A near  the  pump 
house,  run  down  the  slope  to  station  4,  and  then  turn  to  the 
west  along  the  upper  lift  gangway.  The  long  pillars  seen  in 
each  gangway  separate  it  from  the  air  way.  In  every  fifth 
breast  is  written  the  number  by  which  it  is  known. 

Extended  surface  surveys  in  the  mining  regions  come  under 
the  head  of  topography  taken  with  especial  reference  to  geo- 
logic features.  Fig.  56  shows  a small  area  near  Carbondale, 
Pa.,  taken  from  Mine  Sheet  No.  XXI  of  Part  IV  of  the  Atlas  of 


MINE  SURVEYING. 


123 


the  Northern  Anthracite  Coal  Field,  issued  by  the  Second  Geo- 
logical Survey  of  Pennsylvania.  The  scale  is  1 inch  to  800  feet 
and  the  contour  interval  is  10  feet,  the  elevations  being  given 
with  reference  to  tide  water.  The  coordinate  lines,  drawn  at 
intervals  of  2000  feet,  give  distances  north  and  east  from  a 


monument  in  the  yard  of  the  court-house  at  Wilkes  Barre. 
Bore-holes,  dips  of  strata,  and  outcrops  of  the  formations  aie 
shown,  as  also  property  lines,  and  names  of  owners  or  lessees. 
The  colors  on  the  original  map  are  not  reproduced  in  the 
copy. 


124 


TOPOGRAPHIC  SURVEYING. 


Prob.  29.  By  surveys  and  computations  the  following  data 
were  obtained  concerning  four  points  in  a certain  gangway 
driven  around  one  end  of  a vein  in 
@n"  a coal  basin: 

Station.  Latitude.  Longitude. 

A + 2604.25  -+  2428.10 

B + 2597.18  + 2010.43 

N + 3345.65  + 2904.18 

Also,  elevation  of  A = 783.84,  ele- 
vation of  N — 807.90,  azimuth  of 
— MN  - 92°  17'  (S  87°  43'  E).  It  is 
r ‘desired  to  drive  a tunnel  from  A to 
N,  and  for  this  purpose  the  follow- 
ing quantities  are  required  to  be 
found  : (1)  Length  of  line  AN,  (2)  azimuth  of  AN,  (3)  the 
horizontal  angle  BAN,  (4)  the  horizontal  angle  MNA,  (5)  the 
grade  of  the  line  AN. 

Art.  40.  The  True  Meridian. 

A true  meridian  is  established  by  actually  staking  out  a lin a 
running  due  north  and  south,  or  by  determining  the  true  azi  - 
muth  of  a given  line.  The  latter  method  is  preferable  in  town 
and  city  work.  From  the  azimuth  found  for  the  one  line  the 
azimuths  of  all  other  important  lines  are  obtained  by  travers- 
ing or  by  triangulation.  A meridian  actually  staked  out  is  cf 
no  value  except  for  determining  the  azimuths  of  lines.  Three 
methods  of  determining  the  true  meridian  will  be  here  ex- 
plained. 

By  Polaris  and  Mizar. — The  pole-star  Polaris  revolves 
around  the  pole  in  a small  circle,  and  crosses  the  meridian,  or 
culminates,  twice  each  day.  Mizar,  the  middle  one  of  the  three 
stars  in  the  tail  of  the  Great  Bear  or  handle  of  the  Great 
Dipper,  revolves  around  the  pole  in  a large  circle  and  culmi- 
nates a few  minutes  earlier  than  Polaris.  In  1895  Polaris  cul- 
minates about  50  seconds  after  it  and  Mizar  are  in  the  same 
vertical  circle,  in  1900  about  2£  minutes  after,  and  in  1905 
about  4£  minutes  after,  the  annual  increase  being  21  seconds. 
To  obtain  the  true  meridian  set  up  a transit  about  a quarter  of 
an  hour  before  the  two  stars  are  in  the  same  vertical ; the 


THE  TRUE  MERIDIAN. 


125 


transit  must  be  in  good  adjustment,  particularly  in  respect  to 
collimation  and  horizontal  axis  of  the  telescope.  Sight  alter- 
nately upon  Polaris  and  Mizar,  and  note  by  a watch  the  time 
when  they  are  upon  the  same  vertical.  Then,  after  the  expi- 
ration of  the  interval  above  mentioned,  turn  the  vertical  hair 
upon  Polaris,  and  the  line  of  sight  coincides  with  the  true 
meridian.  The  error  of  this  method  will  probably  be  greater 
than  one  minute  of  angle,  as  the  work  must  be  done  at  night. 

By  Polaris. — The  time  of  culmination  of  Polaris  may  be  as- 
certained from  Table  V,  and  the  vertical  hair  of  a transit  be 
set  upon  it  at  that  instant.  But  a more  accurate  method  is  to 
observe  Polaris  at  its  east  or  west  elongation,  following  it  with 
1 he  vertical  hair  until  its  motion  in  azimuth  ceases.  The  ap- 
] iroximate  time  of  elongation  may  be  found  from  Table  V,  and 
1,  be  astronomical  azimuth  of  Polaris  at  elongation  is  found  from 
Table  YI.  Thus  the  azimuth  of  the  line  of  sight  is  known  ; if 
j!',  point  be  marked  beneath  the  plumb-bob  and  another  several 
hundred  feet  away  in  the  line  of  sight,  a line  is  determined 
whose  azimuth  is  known.  By  repeating  the  operation  on  sev- 
e ral  days  a mean  result  can  be  obtained  which  can  be  depended 
upon  with  an  error  not  exceeding  one  minute  of  angle.  This 
work  need  not  be  done  at  night,  as  Polaris  can  often  be  seen 
by  a telescope  of  moderate  power  in  the  daytime. 

By  the  Sun. — With  a transit  having  a solar  attachment  the 
true  meridian  can  be  found  by  observing  the  sun  at  any  time 
except  between  11  a.m.  and  1 p.m.  Such  an  attachment  can  be 
placed  upon  any  transit  at  a cost  of  about  fifty  dollars.  Accom- 
panying it  is  a pamphlet  giving  full  directions  for  use  and 
adjustment,  together  with  tables  of  the  declination  of  the  sun 
for  Greenwich  noon  on  each  day  of  the  year.  Both  the  transit 
and  the  solar  attachment  should  be  in' correct  adjustment  in  or- 
der to  do  good  work  in  determining  the  true  meridian. 

In  order  to  explain  the  theory  of  the  solar  attachment  let  the 
upper  part  of  Fig.  58  be  a section  of  the  celestial  sphere  in  the 
plane  of  the  true  meridian,  W and  S being  the  north  and  south 
points  of  the  horizon,  P the  pole,  Z the  zenith,  Q the  celes- 
tial equator,  and  0 the  place  of  the  sun  at  noon.  Let  A be 
the  point  where  the  instrument  is  set,  which  may  be  regarded 


126 


TOPOGRAPHIC  SURVEYING. 


as  tlie  center  of  the  celestial  sphere.  Then  the  angle  PAN  or 
its  equal  QAZ  is  the  latitude  of  the  place  of  observation.  The 
angle  QAO  is  the  declination  of  the 
sun,  which  is  positive  when  the 
sun  is  north  of  the  equator  from 
March  21  to  September  21,  and  neg- 
ative when  the  sun  is  south  of  the 
equator  from  September  21  to 
March  21.  The  lower  part  of  Fig. 
58  is  a plan,  A being  the  place  of 
the  instrument,  NS  the  true  me- 
ridian through  A,  W and  E the 
west  and  east  directions,  AO  the  direction  of  the  sun  about 
10  o’clock  in  the  morning,  and  AL  a line  whose  azimuth  is 
required  to  be  found. 


Fig.  58. 


Let  db  represent  the  telescope  of  the  transit,  placed  in  the 
meridian  and  elevated  so  as  to  point  to  the  celestial  equator ; 
this  will  be  the  case  when  the  angle  of  elevation  SAQ  is  equal 
to  the  co-latitude,  or  when  SAQ  — 90°  — QAZ.  Let  cd  be 
the  telescope  of  the  solar  attachment  pointing  toward  the  sun; 
then  the  vertical  angle  between  db  and  cd  is  equal  to  the  dec- 
lination of  the  sun  QAO.  In  this  position  the  solar  attach- 
ment is  like  an  equatorial  telescope,  its  axis  pointing  to  the 
pole  P,  and  as  the  sun  moves  the  telescope  cd  will  follow  it 
along  the  celestial  sphere  until  the  change  in  declination  be- 
comes appreciable. 


Before  beginning  work  a list  of  hourly  declination  settings 
is  to  be  prepared  by  help  of  the  table  of  declinations  which  is 
furnished  by  the  maker  of  the  instrument.  This  table  also 
gives  for  each  hour  the  effect  of  refraction,  this  refraction  al- 
ways increasing  the  altitude  of  the  sun.  For  example,  let  it 
be  required  to  find  the  declination  settings  for  the  afternoon  of 
September  19,  1895,  for  any  place  where  eastern  standard  time 
is  used.  The  table  gives  -J-  1°  28'  54"  as  the  declination  of  the 
sun  at  Greenwich  noon  for  that  day,  and  58"  as  the  hourly  de- 
crease of  declination.  The  declination  at  7 a.m.  of  eastern 
standard  time  is  then  +1°  28'  54",  and  that  at  5 r.M.  is 
+ 1°  28'  54"  - 10  X 58"  = + 1°  21'  14".  Thus  the  declination 


THE  TRUE  MERIDIAN", 


127 


for  each  hour  is  found  and  given  in  the  second  column.  In  the 
third  column  is  placed  the  refraction  correction  as  given  in  the 
table,  and  the  fourth  column  gives  the  final  declination  settings 


Hour. 

Declination. 

Refraction 

Correction. 

Declination 

Settings. 

Remarks. 

1 P.M. 

2 P.M. 

3 P.M. 

4 pm. 

5 pm. 

-f  1°  25'  06" 
+ 1 24  08 
4-  1 23  10 
+ 1 22  12 
-f  1 21  14 

+ O'  48" 
4-0  5 4 
-f  1 05 
-FT  32 
4 2 51 

+ 1°  25'  54" 
+ 1 24  52 
+ 1 24  15 
■ 4-1  23  44 
+ 1 23  05 

For  Eastern 
Standard  Time, 
September  19, 
1895. 

which  are  the  apparent  declinations  for  the  respective  hours. 
The  refraction  correction  is  always  additive,  and  hence  if  the 
declination  is  south  or  negative  its  numerical  value  is  decreased, 


Hour. 

Declination. 

Refraction 

Correction. 

Declination 

Settings. 

Remarks. 

8 a.m. 

9 A M. 

10  A.M. 

11  A.M. 

— 22°  23'  43" 

— 22  24  02 

— 22  24  21 

— 22  24  40 

4-  6'  31" 
4-  2 59 
4 2 11 
+ 1 54 

— 22°  17'  12" 

— 22  21  03 

— 22  22  10 
— 22  22  46 

For  Eastern 
Standard  Time, 
December  5, 
1895. 

as  the  example  for  December  5,  1895,  shows  ; on  that  day  the 
table  gives  the  declination  at  Greenwich  noon  as  22°  23'  24" 
south  and  the  hourly  change  as  19  seconds. 

After  this  list  is  made  out  the  observer  sets  up  the  transit 
over  the  point  A in  order  to  find  the  true  azimuth  of  a line  AL 
(Fig.  58).  The  telescope  is  leveled  by  the  attached  bubble  and 
pointed  approximately  toward  the  south.  The  declination  set- 
ting for  the  hour  is  next  laid  off  on  the  vertical  arc,  depress- 
ing the  object  glass  if  the  declination  is  positive  and  elevating 
it  if  the  declination  is  negative.  The  telescope  of  the  solar  is 
then  leveled  by  means  of  its  own  bubble,  and  thus  the  angle 
between  the  two  telescopes  is  the  same  as  the  apparent  decli- 
nation of  the  sun  QAO.  Both  telescopes  are  then  elevated  until 
the  vertical  arc  reads  an  angle  equal  to  the  co-latitude  of  the 
place,  or  The  solar  attachment  is  next  turned  on  its  axis, 

and  the  limb  of  the  transit  upon  its  axis,  until  the  sun  is  seen 
inscribed  in  the  square  formed  by  the  four  extreme  cross-hairs 


128 


TOPOGRAPHIC  SURVEYING. 


in  the  focus  of  the  solar  telescope.  When  this  is  the  case,  the 
transit  telescope  is  in  the  plane  of  the  meridian,  and, if  desired 
a point  may  he  set  out  in  the  line  AS  to  mark  that  meridian. 

It  will  be  better,  however,  to  read  both  verniers  on  the  hori- 
zontal circle,  then  turn  the  alidade  around  to  L and  read  both 


Tinje. 

Reading  on 
Meridian. 

A.  B. 

Reading  on  L. 
A.  B. 

Angle  SAL. 

Remarks. 

9:15  a.m. 

20°  19' 

00" 

30" 

182°27/ 

30" 

30" 

162°  08'  15" 

Oct.  28,  1885 

9:30 

80  00 

15 

15 

242  08 

30 

00 

162  09  00 

R.  Doe, 

9:45 

140  59 

30 

15 

303  08 

45 

15 

162  09  08 

Observer. 

3:15  p.m. 

200  01 

60 

45 

2 09 

45 

30 

162  07  45 

Mean  = 

3:30 

200  12 

45 

30 

62  21 

15 

30 

162  08  45 

162°  08'  38" 

4:00 

320  06 

00 

00 

122  14 

45 

60 

162  08  53 

Azimuth  AL 
= 17°  5P  22" 

verniers  again.  The  angle  SAL,  which  is  the  azimuth  of  L, 
has  thus  been  measured.  Repeating  again  the  operation  with 
the  solar  another  value  of  SAL  is  determined,  and  by  making 
several  measures,  both  in  the  morning  and  afternoon,  the  mean 
result  can  be  relied  upon  with  a probable  error  of  about  one 
minute  if  the  observer  be  skilled  in  such  work.  The  above 
form  indicates  a method  of  keeping  the  field-notes. 

By  an  Altitude  of  the  Sun. — The  altitude  of  the  sun  may  be 
taken  with  a common  transit,  and  this,  together  with  the 
declination  of  the  sun  and  the  latitude  of  the  place,  gives  the 
means  of  computing  tLe  azimuth  of  the  sun  at  the  moment  of 
observation.  This  method  is  explained  in  full  on  page  243. 


detic  $ 


GEODETIC 


Table  I. 

NATURAL  SINES  AND  COSINES 

TO 


FIVE  DECIMAL  PLACES. 


130  TABLE  I.  SINES  AND  COSINES. 


0 

o 

1 

o 

2 

0 

3 

° ! 

4° 

Sine 

Cosin 

Sine 

Cosin 

Sine 

Cosin 

Sine 

Cosin 

Sine 

Cosin 

0 

.00000 

One. 

.01745 

.99985 

.03490 

.99939 

.05234 

799863 

76(5976 

799756 

60 

1 

.00029 

One. 

.01774 

.99984 

.03519 

.99938 

.05263 

.99861! 

.07005 

.99754 

59 

2 

.00058 

One. 

.01803 

.99984 

.03548 

.99937 

.05292 

.99860! 

.07034 

.99752 

58. 

3 

.00087 

One. 

.01832 

.99983 

.03577 

.99936 

.05321 

.99858 

.07063 

.99750 

57 

4 

.00116 

One. 

.01862 

.99983 

.03606 

.99935 

.05350 

.99857 

.07092 

.99748 

56 

5 

.00145 

One. 

.01891 

.99982 

.03635 

.99934 

.053/9 

.99855 

.07121 

.99746 

55 

6 

.00175 

One. 

.01920 

.99982 

.03664 

.99933 

.05408 

.99854 

.07150 

.99744 

51 

7 

.00204 

One. 

.01949 

.99981 

.03693 

.99932 

.05437 

.99852 

.07179 

.99742 

53 

8 

.00233 

One. 

.01978 

.99980 

.03723 

.99931 

.05466 

.99851 

.07v208 

.99740 

52 

9 

.00262 

One. 

.02007 

.99980 

.03752 

.99930 

.05495 

.99849 

.07237 

.99738 

51 

lO 

.00291 

One. 

.02036 

.99979 

.03781 

.99929 

.05524 

.99847 

.07266 

.99736 

50 

11 

.00320 

.99999 

.02065 

.99979 

.03810 

.99927 

.05553 

.99846 

.07295 

.99734 

49 

12 

.00349 

.99999 

.02094 

.99978 

.03839 

.99926 

.05582 

.99844 

.07324 

.99731 

48 

13 

.00378 

.99999 

.02123 

.99977 

.03868 

.99925 

.05611 

.99842 

.07353 

.99729 

47 

14 

.00407 

.99999 

.02152 

.99977 

.03897 

.99924 

.05640 

.99841 

.07382 

.99727 

46 

15 

.00436 

.99999 

.02181 

.99976 

.03926 

.99923 

.05669 

.99839 

.07411 

. 99725 

45 

16 

.00465 

.99999 

.02211 

.99976 

.03955 

.99922 

.05698 

.99838 

.07440 

.99723 

44 

17 

.00495 

.99999 

.02240 

.99975 

.03984 

.99921 

.05727 

.99836 

.07469 

.99721 

43 

18 

.00524 

.99999 

.02269 

.99974 

.04013 

.99919 

.05756 

.99834 

.07498 

.99719 

42 

19 

.00553 

.99998 

.02298 

.99974 

.04042 

.99918 

.05785 

.99833 

.07527 

.99716 

41 

20 

.00582 

.99998 

.02327 

.99973 

.04071 

.99917 

.05814 

.99831 

.07556 

.99714 

40 

21 

.00611 

.99998 

.02356 

.99972 

.04100 

.99916 

.05844 

.99829 

.07585 

.99712 

39 

22 

.00640 

.99998 

.02385 

.99972 

.04129 

.99915 

.05873 

.99827! 

.07614 

.99710 

38 

23 

.00669 

.99998 

.02414 

.99971 

.04159 

.99913 

.05902 

.99826! 

.( 7643 

.99708 

37 

24 

.00698 

.99998 

.02443 

.99970 

.04188 

.99912 

.05931 

.99824| 

.07672 

.99705 

36 

25 

.00727 

.99997 

.02472 

.99969 

.04217 

.99911 

.05960 

.99822 

.07701 

.99703 

35 

26 

.00756 

.99997 

.02501 

.99969 

.04246 

.99910 

.05989 

.99821] 

.07730 

.99701 

! 34 

27 

.00785 

.99997 

.02530 

.99968 

. 04275 

.99909 

.06018 

.99819 

.07759 

.99699 

33 

28 

.00814 

.99997 

.02560 

.99967 

.04304 

.99907 

.06047 

.99817| 

.07788 

.99696 

! 32 

29 

.00844 

.99996 

.02589 

.99966 

.04333 

.99906 

.06076 

.99815! 

.07817 

.99694 

! 31 

30 

.00873 

.99996 

.02618 

.99966 

.04362 

.99905 

.06105 

. 99813 j 

.07846 

.99692 

30 

31 

.00902 

.99996 

.02647 

.99965 

.04391 

.99904 

.06134 

.99812 

.07875 

.99689 

29 

32 

.00931 

.99996 

.02676 

.99964 

.04420 

.99902 

.06163 

.99810! 

.07904 

.99687 

28 

33 

.00960 

.99995 

.02705 

.99963 

.04449 

.99901 

.06192 

.99808 

.07933 

.99685 

27 

34 

.00989 

1.99995 

.02734 

.99963 

.04478 

.99900 

.06221 

.99806! 

.07962 

.99683 

26 

35 

.01018 

.99995 

.02763 

.99962 

.04507 

.99898 

.06250 

.99804 

.07991 

.99680 

25 

36 

.01047 

99995 

.02792 

.99961 

.04536 

.99897 

.06279 

.99803 

.08020 

.99678 

! 24 

37 

.01076 

.99994 

.02821 

.99960 

! . 04565 

.99896 

.06308 

.99801! 

.08049 

.99676 

123 

38 

.01105 

.99994 

.02850 

.99959 

.04594 

.99894 

.06337 

.99799 

.08078 

.99673 

22 

39 

.01134 

.99994 

.02879 

.99959 

.04623 

.99893 

.06366 

.99797; 

.08107 

.99671 

21 

40 

.01164 

.99993 

.02908 

.99958 

.04653 

.99892 

.06395 

.99795 

.08136 

.99668 

20 

41 

.01193 

. 99993 

.02938 

.99957 

.04682 

.99890 

.06424 

. 99793 1 

.08165 

.99666 

19 

42 

.01222 

.99993 

.02967 

.99956 

.04711 

.99889. 

.06453 

.99792 

.08194 

.99664 

18 

43 

.01251 

1.99992 

. .02996 

.99955 

.04740 

.99888 

.06482 

.99790 

.08223 

.99661 

17 

44 

.01280 

.99992 

.03025 

.99954 

.04769 

.99886 

.06511 

.99788; 

.08252 

.99659 

16 

45 

.01309 

.99991 

.03054 

1.99953 

.04798 

.99885 

.06540 

.99786 

.08281 

. 99657 

15 

46 

.01338 

1.99991 

.03083 

.99952 

.04827 

.99883 

.06569 

.99784! 

.08310 

.99654 

14 

47 

.01367 

! .99991 

.03112 

.99952 

.04856 

.99882 

.06598 

.99782! 

.08339 

.99652 

13 

48 

.01396 

.99990 

.03141 

.99951 

.04885 

.99881 

.06627 

.99780 

.08368 

.99649 

12 

49 

.01425 

.99990 

.03170 

.99950 

.04914 

.99879 

.06656 

.99778 

.08397 

.99647 

11 

50 

.01454 

. 99989 

.03199 

.99949 

.04943 

.99878 

.06685 

.99776 

.08426 

.99644 

10 

51 

.01483 

.99989 

.03228 

.99948 

.04972 

.99876 

.06714 

.997741 

.08455 

.99642 

9 

52 

.01513 

.99989 

.03257 

.99947 

.05001 

.99875 

.06743 

.99772 

.08484 

.99(539 

8 

53 

.01542 

'.99988 

.03286 

.99946 

.05030 

.99873 

.06773 

.99770 

.08513 

.99637 

7 

54 

01571 

.99988 

.03316 

.99945 

.05059 

.99872 

.06802 

.99768 

.08542 

.99635 

6 

55 

.01600 

.99987 

.03345 

.99944 

.05088 

.99870 

.06831 

.99766 

.08571 

.99632 

5 

56 

.01629 

.99987 

.03374 

.99943 

.05117 

.99869 

.06860 

.99764 

.08600 

.99630 

4 

57 

.01658 

.99986 

.03403 

.99942 

.05146 

.99867 

.06889 

.99762 

.08629 

.99627 

3 

58 

! .016871 

.99986 

.03432 

.99941 

.05175 

.99866 

.06918 1 

.997(50 

.08(558 

.99(525 

2 

59 

.01716 

.99985! 

.03461 

.99940 

.05205 

.99864 

.06947 

.99758 

.08687 

.99(5221 

1 

60 

.01745 

.999851 

.03490 

.99939 

.05234 

.99863 

.06976 

.99756 

.08716 

.99619 

0 

/ 

Cosin 

Sine  l 

Cosin 

Sine 

Cosin 

Sine 

Cosin  | 

Sine  I 

Cosin 

"Sine 

/ 

8( 

3°  i 

o 

OO 

00 

87*  ! 

86“  ! 

85°  1 

TABLE  I.  SINES  AND  COSINES. 


131 


5 

o 

6 

o 

7 

0 1 

8 

« 

? 

0 1 

Sine 

Cosin 

Sine 

Cosin 

Sine 

Cosin 

Sine 

Cosin 

Sine 

Cosin  1 

f 

0 

.08716 

.99619 

.10453 

.99452 

.12187 

“99255 

“13917 

“99027 

.15643 

.98769! 

60 

1 

.08745 

.99617 

. 10482 

.99449 

.12216 

.99251 

.13946 

.99023 

. 15672 : 

.98764 

59 

2 

.08774 

.99614 

.10511 

.99446 

.12245 

.99248 

.13975 

.99019 

.15701 

.98760 

58 

3 

.08803 

.99612 

. 10540 

.99443 

.12274 

.99244 

.14004 

.99015 

.15730 

.98755 

57 

4 

.08831 

.99609 

.10569 

.99440 

.12302 

.99240, 

.14033 

.99011 

. 15758 

.98751 

56 

5 

.08860 

.99607 

.10597 

.99437 

.12331 

.99237; 

.14061 

.99006 

.15787 

.98746 

55 

6 

.08889 

.99604 

.10626 

.99434 

.12360 

.99233 

.14090 

.99002 

.15816 

.98741 

54 

7 

.08918 

99602 

.10635 

.99431 | 

.12389 

.992301 

.14119 

.98998 

.15845 

.98737  53 

8 

.08947 

.99599 

.10684 

.99428! 

.12418 

.99226) 

.14148 

.98994 

.15873 

. 98732 1 

52 

9 

.08976 

.99596 

.10713 

.99424 

.12447 

.99222 

.14177 

.98990 

.15902 

.98728 

51 

10 

.090051 

.99594 

.10742 

.99421 

.12476 

.99219 

.14205 

.98986 

.15931 

.98723, 

50 

11 

.09034 

.99591 

.10771 

.99418 

.12504 

.99215 

.14234 

.98982 

! .15959 

' .98718 

49 

12 

.09063 

.99588 

.10800 

.99415 

.12533 

.99211 

.14263 

.98978 

! .15988 

.98714 

48 

13 

.09092 

.99586 

.10829 

.99412| 

.12562 

.99208; 

.14292 

.98973 

j .16017 

.98709,  47 

14 

.09121 

.99583 

.10858 

.99409) 

.12591 

.992041 

.14320 

.98969 

. 16046 ! 

.98704 

46 

15 

.09150 

.99580 

.10887 

.99406 j 

.12620 

.99200 

.14349 

.98965 

) .16074 

.98700 

45 

16 

.09179 

.99578 

.10916 

.99402 ! 

.12649 

.99197! 

.14378 

.98961 

! .16103! 

.98695; 

44 

17 

.09208 

.99575 

.10945 

.993991 

.12678 

.99193| 

.14407 

.98957 

) .16132 

.98690| 

43 

18 

.09237 

.99572 

.10973 

.99396 ! 

.12706 

.99189 

.14436 

.98953 

.16160 

.98686 

42 

19 

.09266 

.99570 

.11002 

.993931 

.12735 

.99186! 

. 14464 

.98948 

; .16189 

.98681| 

41 

20 

.09295 

.99567 

.11031 

.99390 

.12764 

.99182 

.14493' 

.98944 

.16218 

.98676 

40 

21 

.09324 

.99564 

.11060 

. 99386 i 

.12793 

.99178 

.14522 

.98940 

! .16246 

.98671 

39 

22 

.09353 

.99562 

.11039 

.99383: 

.12822 

.99175 

.14551 

.98936 

.16275 

.98667; 

38 

23 

.09382 

.99559 

.11118 

.99380, 

.12851 

.99171 

.14580 

.98931 

.16304 

.98662! 

37 

24 

.09411 

.99556 

.11147 

.99377 

.12880 

.99167 

.14608 

.98927 

! .16333 

.98657 

36 

25 

.09440 

. 99553 

.11176 

.99374! 

.12908 

.99163 

.14637 

.98923 

! .16361 

.98652! 

35 

26 

.09469 

.99551 

.11205 

.99370 i 

.12937 

.99160) 

.14666 

.98919 

) .16390 

.98648 

34 

27 

.09498 

.99548 

.11234 

.993671 

.12966 

.99156! 

.14695 

.98914 

.16419 

.98643 

33 

28 

.09527 

.99545 

.11263 

.99364; 

.12995 

.99152! 

.14723 

.98910 

! .16447 

.98638 

32 

29 

.09556 

.99542 

.11291 

.99360! 

.13024 

.99148) 

.14752 

.98906 

.16476 

.98633! 

31 

30 

.09585 

.99540 

.11320 

.99357 

.13053 

.99144 

.14781 

.98902 

. 16505 ) 

.98629) 

30 

31 

.09614 

.99537 

.11349 

.99354 

.13081 

.99141 

.14810 

.98897 

! .16533 i 

.98624' 

29 

32 

.09642 

.99534 

.11378 

.99351 

.13110 

.99137 

.14838 

.98893 

.16562| 

.98619; 

28 

33 

.09671 

.99531 

.11407 

.99347 

.13139 

.99133 

.14867 

.98889 

.16591 

.98614 

27 

34 

.09700 

.99528 

.11436 

.99344 

.13168 

.99129 

.14896 

.98884 

.16620 

.98609 

26 

35 

.09729 

.99526 

.11465 

.99341 

.13197 

.99125 

.14925 

.98880 

.16648 

. 98604 1 

25 

36 

.09758 

.99523 

.11494 

.99337 

.13226 

.99122 

.14954 

.98876 

! .16677 

.98600: 

24 

37 

.09787 

.99520 

.11523 

.99334 

.13254 

.99118 

.14982 

.98871 

) .16706 

j.98595 

23 

38 

.09816 

.99517 

.11552 

.99331 

.13283 

.99114) 

.15011 

.98867 

.16734 

1.98590  22 

39 

.09845 

.99514 

.11580 

1.99327; 

.13312 

.99110) 

.15040 

.98863 

| .16763 

1.98585, 

21 

40 

.09874 

.99511 

.11609 

.99324 

.13341 

.99106 

.15069 

.98858 

.16792 

.98580 

20 

41 

.09903 

.99508 

.11638 

.99320! 

.13370 

.99102 

.15097 

.98854 

.16820 

!. 98575 

19 

42 

.09932 

.99506 

.11667 

.99317! 

.13399 

.99098 

.15126 

.98849 

i .16849 

! . 98570 ; 

IS 

43 

.09961 

.99503 

.11696 

.99314! 

.13427 

.99094 

.15155 

.98845 

.16878 

.98565 

17 

44 

.09990 

.99500 

.11725 

.99310 

.13456 

.99091 

.15184 

.98841 

j .16906 

.98561 

16 

45 

.10019 

.99497 

.11754 

•993071 

.13485 

.99087 

.15212 

.98836 

.16935 

.98556 

15 

46 

.10048 

.99494 

.11783 

.99303 

.13514 

.99083 

.15241 

.98832 

.16964 

!. 98551 

; 14 

47 

.10077 

.99491 

.11812 

.99300 

.13543 

.99079 

.15270 

.98827 

! .16992 

1.98546 

i 13 

48 

.10106 

.99488 

.11840 

.99297 

.13572 

.99075 

.15299 

.98823 

.17021 

.98541 

12 

49 

.10135 

.99485 

.11869 

.99293 

.13600 

.99071 

.15327 

.98818 

.17050 

1.98536 

1 11 

50 

.10164 

.99482 

.11898 

.99290 

.13629 

.99067 

.15356 

.98814 

j .17078 

| . 98531 

! 10 

51 

.10192 

.99479 

.11927 

.99286 

.13658 

.99063 

.15385 

.98809 

.17107 

| .98526 

9 

52 

.10221 

.99476 

.11956 

.99283 

.13687 

.99059 

.15414 

.98805 

.17136 

!. 98521 

8 

53 

. 10250 

.99473 

.11985 

.99279 

.13716 

.99055 

.15442 

.98800 

.17164 

1.98516 

7 

54 

.10279 

.99470 

.12014 

.99276 

.13744 

.99051 

.15471 

.98796 

.17193 

.98511 

6 

55 

.10308 

.99467 

.12043 

.99272 

.13773 

.99047 

.15500 

.98791 

.17222 

.98506 

5 

56 

.10337 

.99464 

i .12071 

.99269j 

.13802 

.99043 

.15529 

.98787 

.17250 

.98501 

4 

57 

10366 

.99461 

! .12100 

.99265 

.13831 

.99039 

.15557 

.98782 

.17279 

.98496 

3 

58 

.10395 

.99458 

.12129 

.99262 

.13860 

.99035 

. 15586 

.98778 

! .17308 

.98491 

2 

59 

.10424 

.99455 

.12158 

.99258! 

.13889 

.99031 

.15615  .98773 

.17336 

;. 98486 

1 

60 

.10453 

.99452 

! .12187 

.99255 

.13917 

.99027 

.15643  .98769 

.17365 

!.98481 

_0 

Cosin 

Sine 

Cosin 

Sine 

Cosin 

Sine 

Cosin 

| Sine 

Cosin 

Sine 

/ 

L 

CO 

83° 

82° 

81° 

I 80J 

TABLE  I.  SINES  AND  COSINES. 


132 


10° 


79° 


11° 


Sine 

Cosin 

“6 

.17365 

.98481 

l 

.17393 

.98476 

2 

.17422 

.98471 

3 

.17451 

.98466 

4 

. 17479 

.98461 

5 

.17508 

.98455 

6 

.17537 

.98450 

7 

. 17565 

.98445 

8 

.17591 

.98440 

9 

.17623 

.98435 

10 

.17651 

.98430 

11 

.17680 

.98425 

12 

. 17708 

.98420 

13 

.17737 

.98414 

14 

.17766 

.98409 

15 

.17794 

.98404 

16 

.17823 

.98399 

17 

.17852 

.98394 

18 

.17880 

.98389 

19 

.17909 

.98383 

20 

.17937 

.98378 

21 

.17966 

.98373 

22 

.17995 

.98368 

23 

.18023 

.98362 

24 

.18052 

.98357 

25 

.18081 

.98352 

26 

.18109 

.98347 

27 

.18138 

.98341 

28 

.18166 

.98336 

29 

.18195 

.98331 

30 

.18224 

.98325 

31 

.18252 

.98320 

32 

.18281 

.98315 

33 

.18309 

.98310 

34 

.18338 

.98304 

35 

.18367 

.98299 

36  ! 

.18395 

.98294 

37  ! 

.18424 

.98288 

38  j 

.18452 

.98283 

39 

.18481 

.98277 

40 

. 18509 

.98272 

! 41  ■ 

.13538 

.98267 

i 42 

.1856? 

.98261 

j 43  . 

.18595 

.98256 

! 44 

.18624 

.98250 

| 45  j 

. 18652 

.98245 

i 46  ! 

. 18681 

.93240 

; 47  i 

.18710 

.98234 

I 48  1 

.18738 

.98229 

49 

. 18767 

.98223 

' 50 

.18795 

.98218 

51 

.18824 

.98212 

| 52 

.18852 

.98207 

' 53 

.18881 

.98201 

1 54 

.18910 

.98196 

j 55 

.18938 

.98190 

1 56 

.18967 

.98185 

j 57 

. 18995 

.98179 

1 58 

. 19024 

.98174 

50 

. 19052 

.98168 

60 

.19081 

(.98163  j 

t 

Cosin 

, Sine  i 

Sine 
.19081 
.19109 
.19138 
.19167 
. 19195 
.19224 
.19252 
.19281 
.19309 
.19338 
.19360 

.19395 

.19423 

.19-152 

.19481 

.19509 

.19538 

.19566 

.19595 

.19623 

.19652 

.19680 

.19709 

.19737 

.19766 

.19794 

.19823 

.19851 

.19880 

.19908 

.19937 

.19965 

.19994 

.20022 

.20051 

.20079 

.20108 

.20136 

.20165 

.20193 

.20222 

.20250 

.20279 

.20307 

.20336 

.20364 

.20393 

.20421 

.20450 

.20478 

.20507 

.20535 

.20563 

.20592 

.20620 

.20649 

.20677 

.20706 

.20734 

.20763 

.20791 

Cosin 


Cosin 
.98163 
.98157 
.98152 
.98146 
.98140 
.98135 
.98129 
.98124 
.98118 
.98112 
. 98107  J 

.98101 

.98096 

.98090 


98079 

98073 

98067 

98061 


98050  j 

98044 ! 
98039j 
93033 : 
93027  j 
93021 
98016 | 
98010 
98004 ! 
97993 
97992  j 

97987 
97981 | 
97975 1 
97969 ! 
97963 1 
97958 
97952 , 
97946 1 
97940  j 
97934 

97928j 
97922 
97916 
97910 | 
97905 
97899 
97893 
97887 
97881 
97875 

.97869 

.97863 

.97857 

.97851 

.97845 

.97839 

.97833 

.97827 

.97821 

.97815 


12° 


J3in<3 

720791 

.20820 

.20848 

.20877 

.20905 

.20933 

.20962 

.20990 

.21019 

.21047 

.21076 

.21104 

.21132 

.21161 

.21189 

.21218 

.21246 

.21275 

.21303 

.21331 

.21360 

.21388 

.21417 

.21445 

.21474 

.21502 

.21530 

.21559 

.21587 

.21616 

.21644 

.21672 

.21701 

.21729 

.21758 

.21786 

.21814 

.21843 

.21871 

.21899 

.21928 

.21956 

.21985 

.22013 

.22041 

.22070 

.22098 

.22126 

.22155 

.22183 

.22212 

.22240 

.22268 

.22297 

.22325 

.22353 

.22382 

.22410 

.22438 

.22467 

.22495 


Cosin 


Sine 


78° 


Cosin 


.97815; 
.97809 
. 97803 j 
.977971 
.97791 
.97784| 
.97778 
.97772; 
.97766! 
.97760 
.97754; 

.97748 
.97742 
.97735 
.97729 
.97723| 
.977171 
.97711; 
.97705 
.97698: 
. 97692 1 

.97686 

.97680, 

.97673 

.97667! 

.97661 

.97655 

.97648! 

.97642! 

.97636! 

.97630 

.97623 

.97617 

.97611 

.97604 

.97598 

.97592 

.97585 

.97579 

.97573 

.97566 

.97560 

.97553 

.97547 

.97541 

.97534 

.97528 

.97521 

.97515 

.97508 

.97502 

.97496 

.97489 

.97483 

.97476! 

.97470 

.97463 

.97457 

.97450 

.97444! 

.97437 

Sine 


77° 


13° 

14° 

7 

Sine 

Cosin 

Sine 

Cosin 

.22495 

.97437 

.24192 

797030 

60 

.22523 

.97430 

.24220 

.97023 

59 

.22552 

.97424 

.24249 

.97015 

58 

.22580 

.97417 

.24277 

.97008 

57 

.22608 

.97411 

.24305 

.97001 

j 56 

.22637 

.97404 

.24333 

.96991 

’ 55 

.22665 

*07398 

.24362 

.96987 

i 54 

.22693 

.97391 

.24390 

.96980 

! 53 

.22722 

.97384 

.24418 

.96973 

! 52 

.22750 

.97378 

.24446 

.96966 

51 

.22778 

.97371 

.24474 

.96959 

50 

.22807 

.97365 

.24503 

.96952 

49 

.22835 

.97358 

.24531 

.96945 

48 

.22863 

.97351 

.24559 

.96937 

! 47 

.22892 

.97345 

.24587 

.96930 

! 46 

.22920 

.97338 

.24615 

.96923 

! 45 

.22948 

.97331 

.24644 

.96916 

! 44 

.22977 

.97325 

.24672 

.96909 

! 43 

.23005 

.97318 

.24700 

.96902 

42 

.23033 

.97311 

.24728 

.96894 

41 

.23062 

.97304 

.24756 

.96887 

40 

.23090 

.97298 

.24784 

.96880 

39 

.23118 

. 97291 j 

.24813 

.96873 

! 38 

.23146 

. 97284 

.24841 

.96866 

i 37 

.23175 

.97278 

.24869 

.96858 

i 36 

.23203 

.97271 

.24897 

.96851 

! 35 

.23231 

.97264 

.24925 

.96844 

34 

.23260 

.97257 

.24954 

.96837 

33 

.23288 

.97251 

.24982 

.96829 

32 

.23316 

.97244 

.25010 

.96822 

31 

.23345 

.97237 

.25038 

.96815! 

30 

.23373 

.97230 

.25066 

.96807! 

29 

.23401 

.97223 

.25094 

.96800 

28 

.23429 

.97217 

.25122 

.96793| 

27 

.23458 

.97210 

.25151 

. 96786 ! 

26 

.23486- 

.97203 

.25179 

.96778! 

25 

.23514 

.97196 

.25207 

.96771 

24 

.23542 

.97189 

.25235 

.96764’ 

23 

.23571 

.97182 

.25263 

.96756! 

22 

.23599 

.97176; 

.25291 

.96749! 

21 

.23627 

.97169' 

.25320 

. 96742 ; 

20 

.23656 

.97162’ 

.25348 

. 96734 1 

19 

.23684 

.97155; 

.25376 

.96727 

IS 

.23712 

.97148! 

! .25404 

.96719 

17  i 

.23740 

.971411 

.25432 

.96712 

16  i 

.23769 

.97134! 

.25460 

.96705 

15  ; 

.23797 

.97127 

.25488 

.96697 

14  : 

.23825 

.97120 

.25516 

.96690; 

13  | 

.23853 

.97113 

.25545 

.96682 

12  1 

.23882 

.97106 

.25573| 

. 96675 

11 

.23910 

.97100 

.25601 

.96667; 

10  ; 

.23938 

.97093 

.25629 

.96660 

9 

.23966 

.97086! 

.25657 

.96653 

8 

.23995 

.97079! 

.25685 

.96645 

7 

.24023 

.97072! 

.25713; 

.96638 

6 

.24051 

.97065, 

.25741 

.96630 

5 

.24079 

.97058 

.25769 

.96623 

4 

.24108 

.97051 

.25798 

.96615: 

3 

.24136 

.97044 

.25826 

.96608 

2 

.24164 

.97037 

.25854 

.96600! 

1 

.24192 

.97030 

.25882 

.96593 

0 

Cosin 

Sine 

Cosin 

Sine 

/ 

76°  ! 

75° 

• — ■ r 

table  i.  sines  and  cosines. 


133 


15°  1 

16°  1 

17°  i 

18° 

19° 

Sine 

Cosin 

Sine 

Cosin ; 

Sine 

Cosin 

Sine 

Cosin 

Sine 

Cosin 

0 

.25882 

.96593 

.27564 

.96126 

.29237 

795630 

“30902 

.95106 

.32557 

'94552 

60 

1 

.25910 

.96585 

.27592 

.96118 

.29265 

.95622 

.30929 

.95CJ7 

.32584 

.94542 

! 59 

2 

.25938 

. 96578  j 

.27620 

.96110; 

.29293 

.956131 

.30957 

.95088 

.32612 

.94533 

] 58 

3 

.25966 

.965701 

.27648 

.96102J 

.29321 

.95605 

.30985 

.95079 

! .32639 

.94523 

57 

4 

.25994 

.96562 

.27676 

.96094 

.29348 

.95596; 

.31012 

.95070 

.32667 

.94514 

56 

5 

.26022 

.965551 

.27704 

.96086 

.29376 

. 95588 , 

.31040 

.95061 

.32694 

.94504 

55 

6 

.26050 

.965471 

.27731 

.96078 

.29404 

.95579 

.31068 

.95052 

.32722 

.94495 

54 

7 

.26079 

.96540 

.27759 

* 96070 

.29432 

.95571 

.31095 

.95043 

.32749 

.94485 

53 

8 

.26107 

.965321 

.27787 

.96062 

.29460 

.95562 

| .31123 

.95033 

.32777 

.94476 

52 

9 

.26135 

.965241 

.27815 

.96054! 

.29487 

.95554 

.31151 

.95024 

.32804 

.94466 

51 

10 

.26163 

.96517 

.27843 

.96046 

.29515 

. 95545 

.31178 

.95015 

.32832 

.94457 

50 

11 

.26191 

.96509 

.27871 

.96037 

.29543 

.95536 

! .31206 

.95006 

.32859 

.94447 

49 

12 

.26219 

.965021 

.27899 

.96029 

.29571 

.95528] 

.31233 

.94997 

il  .32887 

.94438 

48 

13 

.26247 

.964941 

.27927 

.96021 

.29599 

.95519 

.31261 

.94988 

.32914 

.94428 

47 

14 

.26275 

.96486! 

.27955 

.96013 

.29626 

.9551l| 

! .31289 

.94979 

.32942 

.94418 

46 

15 

.26303 

. 96479 i 

.27983 

.96005 

| .29654 

.95502; 

; .31316 

.94970 

.32969 

.94409 

45  1 

16 

.26331 

.96471 

.28011 

.95997 

| .29682 

.95493! 

! .31344 

.94961 

.32997 

.94399 

44  I 

17 

.26359 

.96463; 

.28039 

.95989 

.29710 

.95485 

] .31372 

.94952 

.33024 

.94390 

! 43  | 

18 

.26387 

.96456' 

.28067 

.95981 

.29737 

.95476 

.31399 

.94943 

.33051 

.94380 

; 42 

19 

.26415 

.96448 

.28095 

.95972 

| .29765 

.954671 

1 .31427 

.94933 

I .33079 

.94370 

Ui 

20 

.26443 

.96440 

.28123 

.95964 

1 .29793 

.95459 

.31454 

.94924 

.33106 

.94361 

40  j 

21 

.26471 

.96433 

.28150 

.959561 

i .29821 

.95450 

| .31482 

.94915 

i .33134 

.94351 

39 

22 

.26500 

.96425 

.28178 

.95948; 

1,29849 

.95441] 

.31510 

.94906 

.33161 

.94312 

38 

23 

.26528 

.96417 

.28206 

.95940 

.29876 

,95433| 

[ .31537 

.94897 

.33189 

.94332 

37 

24 

. 26556 

.96410 

.28234 

.95931 

.29904 

.95424 

.31565 

.94888 

! .33216 

.94322 

36 

25 

.26584 

.96402 

.28262 

.95923 

.29932 

.95415 

! .31593 

.94878 

] .33244 

.94313 

35 

26 

.26612 

.96394 

.28290 

.95915 

.29960 

.95407' 

! .31620 

.94869 

.33271 

.94303 

31 

27 

.26640 

.96386 

.28318 

.95907 

.29987 

.95398! 

.31648 

.94860 

.33298 

.94293 

33 

28 

.26668 

.96379 

.28346 

.95898 

.30015 

.95389! 

.31675 

.94851 

1 .33326 

.94284 

32 

29 

.26696 

.96371 

.28374 

.95890 

.30043 

.95380] 

.31703 

.94842 

.33353 

.94274 

31 

30 

.26724 

.96363 

.28402 

.95882 

.30071 

.95372 

.31730 

.94832 

.33381 

.94264 

30 

31 

.26752 

.96355 

.28429 

.95874 

.30098 

.95363! 

.31758 

.94823 

1 .33408 

.94254 

29 

32 

.26780 

.96347 

.28457 

.95865 

.30126 

.95354 

.31786 

.94814 

I .33436 

.94245 

28 

33 

.26808 

.96340 

.28485 

.95857 

.30154 

.95345 

.31813 

.94805 

] -33463 

.94235 

27 

34 

.26836 

.96332 

.28513 

.95849 

.30182 

.95337 

.31841 

.94795 

! .33490 

.94225 

26 

35 

.26864 

.96324 

.28541 

.95841 

.30209 

.95328 

.31868 

.94786 

! .33518 

.94215 

25 

S6 

.26892 

.96316 

.28569 

.95832 

.30237 

.95319 

.31896 

.94777 

1 .33545 

.94206 

24 

37 

.26920 

.96308! 

.28597 

.95824 

.30265 

.95310 

.31923 

.94768 

! .33573 

.94196 

23 

38 

.26948 

.96301; 

.28625 

.95816 

.30292 

.95301 

.31951 

.94758 

! .33600 

.94186 

23 

39 

.26976 

.96293 

.28652 

.95807 

.30320 

.95293 

.31979 

.94749 

| .33627 

.94176 

21 

40 

.27004 

.96285; 

.28680 

.95799 

.30348 

.95284 

.32006 

.94740 

.33655 

.94167 

20 

41 

.27032 

.96277 

.28708 

.95791 

.30376 

.95275 

.32034 

.94730 

.33682 

.94157 

19 

42 

.27060 

.96269; 

.28736 

.95782 

.30403 

.95266 

.32061 

.94721 

.33710 

.94147 

18 

43 

.27088 

.96261! 

.28764 

.95774 

.30431 

.95257 

.32089 

.94712 

] .33737 

.94137 

17 

44 

.27116 

.96253 

.28792 

. 95766 

.30459 

.95248 

.32116 

.94702 

.33764 

.94127 

16 

45 

.27144 

.96246 

.28820 

.95757 

.30486 

.95240 

.32144 

.94693 

.33792 

.94118 

15 

46 

.27172 

.96238 

28847 

.95749 

.30514 

.95231 

.32171 

.94684 

.33819 

.94108 

14 

47 

.27200 

.96230 

.*28875 

.95740 

.30542 

.95222 

.32199 

.94674 

.33846 

.94098 

13 

48 

.27228 

.96222 

.28903 

.95732 

.30570 

.95213 

.32227 

.94665 

.33874 

.94088 

12 

49 

.27256 

.962141 

.28931 

.95724 

.30597 

.95204! 

.32254 

.94656 

i 33901 

.94078 

11 

50 

.27284 

.96206 

.28959 

.95715 

.30625 

.95195 

.32282 

.94646 

,.33929 

.94068 

10 

51 

.27312 

.96198 

.28987 

.95707 

! .30653 

.95186 

.32309 

.94637 

! .33956 

.94058 

9 

52 

.27340 

.96190 

.29015 

.95698 

I .30680 

.95177 

.32337 

.94627 

; .33983 

.94049 

8 

53 

.27368 

.96182 

.29042 

.95690 

.30708 

.95168; 

.32364 

.94618 

.34011 

.94039 

7 

54 

.27396 

.96174 

.29070) 

.95681 

.30736 

.95159! 

.32392 

.94609 

.34038 

.94029 

6 l 

55 

.27424 

.96166 

. 29098 i 

. 95673 

.30763 

. 95150 i 

.32419 

.94599 

.34065 

.94019 

5 ! 

56 

.27452 

.96158 

.29126; 

.95664 

.30791 

.95142 

.32447 

.94590 

.34093 

.94009 

4 i 

57 

.27480 

.96150 

.29154, 

. 95656 

.30819 

.95133 

.32474 

.93580 

.34120 

.93999 

3 ! 

58 

.27508 

.96142 

.29182 

.95647 

.30846 

.95124 

.32502 

.94571 

.34147 

.93989 

2 

59 

.27536 

.96134 

.29209! 

.95639 

.30874 

.95115 

| .32529 

.94561 

.34175 

.93979 

1 

60 

.27564 

.96126 

.29237 

. 95630 

.30902 

.95106 

.32557 

.94552 

.34202 

.93969 

0 

Cosin 

Sine 

Cosin  1 

Sine 

Cosin 

Sine  i 

Cosin 

Sine 

j Cosin  j 

Sine 

i 

74° 

73“  1 

72°  1 

71° 

o 

© 

S> 

L-ii 

134 


TABLE  I.  SINES  AND  COSINES. 


20° 

21° 

22° 

23° 

24°  I 

1 

Sine 

Cosin 

Sine 

Cosin 

Sine 

Cosin 

Sine 

Cosin 

Sine  Cosin 

0 

.34202 

.93969 

.35837 

.93358 

.37461 

792718 

.39073 

.92050 

.40674 

.91355; 

60 

1 

.34229 

.93959 

.35864 

.93348 

.37488 

.92707 

.39100 

.92039 

.40700 

.91343, 

50 

2 

.34257 

.93949 

.35891 

.93337 

.37515 

.92697 

.39127 

.92028 

.40727 

.91331 | 

58 

3 

.34284 

.93939 

.35918 

.93327 

.37542 

.92686 

.39153 

.92016 

.40753' 

.91319 

5? 

4 

.34311 

.93929 

.35945 

.93316 

.37569 

.92675 

.39180 

.92005 

.40780 

.91307: 

56 

5 

.34339 

.93919 

.35973 

.933061 

.37595 

.92664 

.39207 

.91994 

.40806! 

.91295! 

55 

6 

.34366 

.93909 

.36000 

.93295' 

.37622 

.92653 

.39234 

.91982 

.40833 

.91283 

54 

7 

.34393 

.93899 

.36027 

.93285 

.37649 

.92642 

.39260 

.91971 

.40860 

.91272 

53 

8 

.34421 

.93889 

.36054 

.93274 

.37676 

.92631 

.39287 

.91959 

.40886 

.91260 

52 

9 

.34448 

.93879 

.36081 

.93264 

.37703 

.92620 

.39314 

.91948 

.40913 

.91248 

51 

10 

.34475 

.93869 

.36108 

.93253 

.37730 

.92609 

.39341 

.91936 

.40939 

.91236, 

50 

11 

.34503 

.93859 

.36i35 

.93243 

.37757 

.92598 

.39367 

.91925 

.40966 

.91224* 

49 

12 

.34530 

.93849 

.36162 

.93232 

.37784 

.92587 

.39394 

.91914 

.40992 

.91212 

48 

13 

.34557 

.93839 

.36190 

.93222 

.37811 

.92576 

.39421 

.91902 

.41019 

.91200 

47 

14 

.34584 

.93829 

.36217 

.93211 

.37838 

.92565 

.39448 

.91891 

.41045 

.91188; 

46 

15 

.34612 

.93819 

.36244 

.93201 

.37865 

.92554 

.39474 

.91879 

.41072 

.91176 

46 

! 16 

.34639 

.93809 

.36271 

.93190 

.37892 

.92543 

.39501 

.91868 

.41098 

.91164  1 

! 44 

17 

.34666 

.93799 

.36298 

.93180 

.3? 919 

.92532 

.39528 

.91856 

.41125 

.91152 

43 

18 

.34694 

.93789 

.36325 

.93169 

.37946 

.92521 

.39555 

.91845 

.41151 

.91140! 

42 

19 

.34721 

.93779 

.36352 

.93159 

.37973 

.92510 

.39581 

.91833 

.41178 

.91128! 

41 

20 

.34748 

.93769 

.36379 

.93148 

.37999 

.92499 

.39608 

.91822 

.41204 

.91116 

I40 

21 

.34775 

.93750 

.36406 

.93137 

.38026 

.92488 

.39635 

.91810 

.41231 

.91104 

! 39 

22 

.34803 

.93748 

.36434 

.93127 

.38053 

.92477 

.39661 

.91799 

.41257 

.91092 

38 

23 

.34830 

.93738 

.36461 

.93116 

.38080 

.92466 

.39688 

.91787 

.41284 

.91080 

37 

24 

.34857 

.93728 

.36488 

.93106 

.38107 

.92455 

.39715 

.91775 

.41310 

.91068 

36 

25 

.34884 

.93718 

.36515 

.93095 

.38134 

.92444 

.39741 

.91764 

.41337 

.91056 

| 35 

26 

.34912 

.93708 

.36542 

.93084 

.38161 

.92432 

.39768 

.91752 

.41363 

.91044 

34 

27 

.34939 

.93698 

.36569 

.93074 

.38188 

.92421 

.39795 

.91741 

.41390 

.91032 

33 

28 

.34966 

.93688 

.36596 

.93063 

.38215 

.92410 

.39822 

.91729 

.41416 

.91020 

32 

29 

.34993 

.93677 

.36623 

.93052 

.38241 

.92399 

.39848 

.91718 

.41443 

.91008 

31 

30 

! .35021 

.93667 

.36650 

.93042 

.38268 

.92388 

.39875 

.91706 

.41469 

.90996 

30 

31 

.35048 

.9365? 

.36677 

.93031 

.38295 

.92377 

.39902 

.91694 

.41496 

.90984 

29 

32 

| .35075 

.93647 

.36704 

.93020 

.38322 

.92366 

.39928 

.91688 

.41522 

.90972 

28 

33 

| .35102 

.93637 

.36731 

.93010 

.38349 

.92355 

.39955 

.91671 

.41549 

.90960 

27 

34 

i .35130 

.93626 

.36758 

.92999 

.38376 

.92343 

.39982 

.91660 

.41575 

.90948 

26 

35 

.35157 

.93616 

.36785 

.92988 

.38403 

.92332 

.40008 

.91648 

i .41602 

.90936 

25 

36 

.35184 

.93606 

.36812 

.92978 

.38430 

.92321 

.40035 

.91636 

.41628 

.90924 

1 24 

37 

.35211 

93596 

.36839 

.92967 

.38456 

.92310 

.40062 

.91625 

.41655 

.90911 

1 23 

38 

.35239 

.93585 

.36867 

.92956! 

.38483 

.92299 

.40088 

.91613 

.41681 

.90899 

! 22 

39 

.35266 

.93575 

.36894 

.92945 

.38510 

.92287 

*40115 

.91601 

.41707 

.90887 

21 

40 

.35293 

.93565 

.36921 

.92935 

.38537 

.92276 

.40141 

.91590 

.41734 

.90875 

20 

41 

.35320 

. 93555 1 

.36948 

.92924 

.38564 

.92265 

.40168 

.91578 

.41760 

.90863 

19 

42 

.35347 

.93544 

.36975 

.92913 

.38591 

.92254 

.40195 

.91566 

.41787 

.90851 

18 

43 

.35375 

.93534 

.37002 

.92902 

.38617 

.92243 

.40221 

.91555 

.41813 

.90839 

i 17 

44 

.35402 

.93524 

.37029 

.92892 

.38644 

.92231 

.40248 

.91513 

.41840 

.90826 

j 16 

45 

.35429 

.93514 

.37056 

,92881 

.38671 

.92220 

.40275 

.91531 

.41866 

.90814 

1 15 

46 

. 35456 

.93503 

.37083 

.92870 

.38698 

.92209 

.40301 

.91519 

.41892 

.90802 

14 

47 

.35484 

1.93493 

.37110 

.92859 

.38725 

.92198 

.40328 

.91508 

.41919 

.90790 

' 13 

48 

.35511 

1.93483 

.37137 

.92849 

.38752 

.92186 

.40355 

.91496 

.41945 

.90778 

12- 

49 

. 35538 

.93472 

.37164 

.92838 

.38778 

.92175 

.40381 

.91484 

.41972 

.90766 

11 

50 

.35565 

,.  93462 

.37191 

.92827 

.38805 

.921641 

.40408 

.91472 

.41998 

.90753 

10 

51 

.35592 

. 93452 1 

.37218 

.92816 

.38832 

.92152 

.40434 

.91461 

.42024 

.90741 

Ci 

52 

.35619 

.93441 

.37245 

.92805 

.38859 

.92141 

.40461 

.91449 

.42051 

.90729 

8 

53 

.35647 

1.93431 

.37272 

.92794 

.38886 

.92130 

.40488 

.91437 

.42077 

.90717 

7 

54 

.35674 

1.93420 

.37299 

.92784 

.38912 

.92119 

.40514 

.91425 

.42104 

.90704 

6 

55 

.35701 

'93410 

. 37326 

.92773 

.38939 

.92107 

.40541 

.91414 

.42130 

.90692 

5 

56 

.35728 

1 .93400 

.37353 

.92762 

| .38966 

J. 92096 

.40567 

.91402 

.42156 

.90680 

4 

57 

.35755 

i. 93389 

.37380 

.92751 

! .38993 

.92085! 

.40594 

.91390 

.42183 

.90668 

g 

58 

.35782 

.93379 

.37407 

.92740 

: . 39020 

.920731 

.40621 

.91378 

.42209 

.90655 

"59 

.35810 

,.93368: 

.37434 

.92729 

.39046 

.92062) 

.40647  .91366 

.42235 

.90643 

1 I 

60 

.35837 

.93358 

.37461 

.92718 

.39073 

.92050: 

i .40674 

.91355 

.422(52 

.90631 

0 

/ 

Cosin 

Sine 

Cosin 

Sine 

Cosin  Sine 

Cosin 

Sine 

Cosin 

Sine 

f 

69° 

68° 

1 67° 

1 66° 

G5° 

TABLE  1.  SINES  AND  COSINES. 


135 


r~ 

25° 

26° 

27° 

to 

00 

o 

29° 

Sine 

Cosin 

Sine 

Cosin 

Sine 

Cosin 

Sine 

Cosin 

Sine 

Cosin 

0 

.42262 

.90631 

.43837 

.89879 

.45399 

789101 

.46947 

.88295 

.48481 

.87462 

60 

l 

.42288 

.90618 

.43863 

.89867 

.45425 

.89087 

.46973 

.88281 

.48506 

.87448 

59 

2 

.42315 

.90606 

.43889 

.89854 

.45451 

.89074 

.46999 

.88267 

.48532 

.87434 

53 

3 

.42341 

.90594 

.43916 

.89841 

.45477 

.89061 

.47024 

.88254 

.48557 

.87420 

57 

4 

.42367 

.90582 

.43942 

.89828 

.45503 

.89048 

.47050 

.88240 

.48583 

.87406 

56 

5 

.42394 

.90569 

.43968 

.89816 

.45529 

.89035 

.47076 

.88226 

.48608 

.87391 

55 

6 

.42420 

.90557 

.43994 

.89803 

.89021 

.47101 

.88213 

.48634 

.87377 

54 

7 

.42446 

.90545 

.44020 

.89790 

.45580 

.89008 

.47127 

.88199 

.48659 

.87363 

53 

8 

.42473 

.90532 

.44046 

.89777 

.45606 

.88995 

.47153 

.88185 

.48684 

.87349 

5,2 

9 

.42499 

.90520 

.44072 

.89764 

.45632 

.88981 

.47178 

.88172 

.48710 

.87335 

51 

10 

.42525 

.90507 

.44098 

.89752 

.45658 

.88968 

.47204 

.88158 

.48735 

.87321 

50 

11 

.42552 

.90495 

.44124 

.89739 

.45684 

.88955 

.47229 

.88144 

.48761 

.87306 

49 

12 

.42578 

.90483 

.44151 

.89726 

.45710 

.88942 

.47255 

.88130 

.48786 

.87292 

43 

13 

.42604 

.90470 

.44177 

.89713 

.45736 

.88928 

.47281 

.88117 

.48811 

.87278 

47 

14 

.42631 

.90458 

.44203 

.89700 

.45762 

.88915 

.47306 

.88103 

.48837 

.87264 

46 

15 

.42657 

.90446 

.44229 

.89687 

.45787 

.88902 

.47332 

.88089 

.48862 

.87250 

45 

16 

.42683 

.90433 

.44255 

.89674 

.45813 

.88888 

.47358 

.88075 

.48888 

.87235 

44 

17 

.42709 

.90421 

.44281 

.89662 

.45839 

.88875 

.47383 

.88062 

.48913 

.87221 

43 

18 

.42736 

.90408 

.44307 

.89649 

.45865 

.88862 

.47409 

.88048 

.48938 

.87207 

42 

19 

.42762 

.90396 

.44333 

.89636 

.45891 

.88848 

.47434 

.88034 

.48964 

.87193 

41 

20 

o 42788 

.90383 

.44359 

.89623 

.45917 

.88835 

.47460 

.88020 

.48989 

.87178 

40 

21 

.42815 

.90371 

.44385 

.89610 

.45942 

.88822 

.47486 

.88006 

.49014 

.87164 

39 

22 

.42841 

.90358 

.44411 

.89597 

.45968 

.88808 

.47511 

.87993 

.49040 

.87150 

38 

23 

.42867 

.90346 

.44437 

.89584 

.45994 

.88795 

.47537 

.87979 

.49065 

.87136 

37 

24 

.42894 

.90334 

.44464 

.89571 

.46020 

.88782 

.47562 

.87965 

.49090 

.87121 

36 

25 

.42920 

.90321 

.44490 

.89558 

.46046 

.88768 

.47588 

.87951 

.49116 

.87107 

35 

26 

.42946 

.90309 

.44516 

.89545 

.46072 

.88755 

.47614 

.87937 

.49141 

.87093 

34 

27 

.42972 

.90296 

.44542 

.89532 

.46097 

.88741 

.47639 

.87923 

.49166 

.87079 

33 

28 

.42999 

.90284 

.44568 

.89519 

.46123 

.88728 

.47665 

.87909 

.49192 

.87064 

32 

29 

.43025 

.9 '271 

.44594 

.89506 

.46149 

.88715 

.47690 

.87896 

.49217 

.87050 

31 

30 

.43051 

.90259 

.44620 

.89493 

.46175 

.88701 

.47716 

.87882 

.49242 

.87036 

30 

31 

.43077 

.90246 

.44646 

.89480 

.46201 

.88688 

.47741 

.87868 

.49268 

.87021 

29 

32 

.43104 

.90233 

.44672 

.89467 

.46226 

.88674 

.47767 

.87854 

.49293 

.87007 

28 

33 

.43130 

.90221 

.44698 

.89454 

.46252 

.88661 

.47793 

.87840 

.49318 

.86993 

27 

34 

.43156 

.90208 

.44724 

.89441 

.46278 

.88647 

.47818 

.87826 

.49344 

.86978 

26 

35 

.43182 

.90196 

.44750 

.89428 

.46304 

.88634 

.47844 

.87812 

.49369 

.86964 

25 

36 

.43209 

.90183 

.44776 

.89415 

.46330 

.88620 

.47869 

.87798 

.49394 

.86949 

24 

37 

.43235 

.90171 

.44802 

.89402 

.46355 

.88607 

.47895 

.87784 

.49419 

.86935 

23 

38 

.43261 

.90158 

.44828 

.89389 

.46381 

.88593 

.47920 

.87770 

.49445 

.86921 

22 

39 

.43287 

.90146 

.44854 

.89376 

.46407 

.88580 

.47946 

.87756 

.49470 

.86906 

21 

40 

.43313 

.90133 

.44880 

.89363 

.46433 

.88566 

.47971 

.87743 

.49495 

.86892 

20 

41 

.43340 

.90120 

.44906 

.89350 

.46458 

.88553 

.47997 

.87729 

.49521 

.86878 

19 

42 

.433,66 

.90108 

.44932 

.89337 

.46484 

.88539 

.48022 

.87715 

.49546 

.86863 

18 

43 

.43392 

.90095 

.44958 

.89324 

.46510 

.88526 

.48048 

.87701 

.49571 

.86849 

17 

44 

.43418 

.90082 

.44984 

.89311 

.46536 

.88512 

.48073 

.87687 

.49596 

.86834 

16 

45 

.43445 

.90070; 

.45010 

.89298 

.46561 

.88499 

.48099 

.87673 

.49622 

.86820 

15 

46 

.43471' 

.90057| 

.45036 

.89285 

.46587 

.88485 

.48124 

.87659 

.49647 

.86805 

14 

47 

.43497 

.90045! 

.45062 

.89272 

.46613 

.88472 

.48150 

.87645 

.49672 

.86791 

13 

48 

.43523 

.90032 

.45088 

.89259 

.46639 

.88458 

.48175 

.87631 

.49697 

.86777 

12 

1 49 

.43549 

.90019| 

.45114 

.89245 

.46664 

.88445 

.48201 

.87617 

.49723 

.86762 

11  i 

i 50 

.43575 

.90007 

.45140 

.89232 

.46690 

.88431 

.48226 

.87603 

.49748 

.86748 

io  ! 

51 

.43602 

.89994 

.45166 

.89219 

.467-16 

.88417 

.48252 

.87589 

.49773 

.86733 

9 

52 

.43628 

.89981 

.45192 

.89208 

.46742 

.88404 

.48277 

.87575 

.49798 

.86719 

8 

53 

.43654 

.89968 

.45218 

.89193 

.46767 

.88390 

.48303 

.87561 

.49824 

.86704 

7 

54 

.43680 

.89956 

.45243 

.89180 

.46793 

.88377 

.48328 

.87546 

.49849 

.86690 

6 

55 

.43706 

.89943 

.45269 

.89167 

.46819 

.88363 

.48354 

.87532 

.49874 

.86675 

5 

56 

.43733 

.89930 

.45295 

.89153 

.46844 

.88349 

.48379 

.87518 

.49899 

.86661 

4 

57 

.43759 

.89918 

.45321 

.89140 

.46870 

.88336 

.48405 

.87504 

.49924 

.86646 

3 

58 

.43785 

.89905 

.45347 

.89127 

.46896 

.88322 

.48430 

.87490 

.49950 

.86632 

2 

59 

.43811 

.89892 

.45373 

.89114 

.46921 

.88308 

.48456 

.87476 

.49975 

.86617 

1 

60 

.43837 

.89879 

.45399 

.89101 

.46947 

.88295 

.48481 

.87462 

.50000 

.86603 

_0 

/ 

Cosin 

Sine 

Cosin 

Sine 

Cosin 

Sine 

Cosin 

Sine 

Cosin 

Sine 

/ 

64° 

63° 

62° 

61° 

o 

O 

CD 

136 


TABLE  T.  SINES  AND  COSINES. 


e 

O 

00 

31° 

o 

CM 

00 

33° 

CO 

o 

Sine 

Cosin 

Sine 

Cosin 

Sine 

Cosin 

Sine 

Cosin 

Sine 

Cosin 

f 

0 

.50000 

.86603 

.51504 

.85717 

.52992 

.84805 

.54464 

.83867 

.55919 

.829041 

60 

1 

.50025 

.86588 

. 51529 

.85702 

.53017 

.847891 

.54488 

.83851 

.55943 

.82887 1 

59 

2 

.50050 

.86573 

.51554 

.85687 

.53041 

.84774! 

.54513 

.83835 

.55968 

.82871 

53 

3 

.50076 

.86559 

.51579 

.85672 

.53066 

.84759! 

.54537 

.83819 

.55992 

.82855:  57 

4 

.50101 

.86544 

.51604 

.85657 

.53091 

.84743 

.54561 

.83804 

.56016 

.82839'  56 

5 

.50126 

.86530 

.51628 

.85642 

.53115 

.84728 

.54586 

.83788 

.56040 

. 82822 j 

55 

6 

.50151 

.86515 

.51653 

-85627 

.53140 

.84712 1 

.54610 

.83772 

.56064 

.82806; 

54 

7 

.50176 

.86501 

.51678 

.85612 

.53164 

.846971 

.54635 

.83756 

.56088 

.82790! 

53 

8 

.50201 

.86486 

.51703 

, 85597 

.53189 

.84681 

.54659 

.83740 

.56112 

.82773; 

52 

9 

.50227 

.86471 

.51728 

.85582 

.53214 

.84666, 

.54683 

.83724 

.56136 

.82757 

51 

10 

.50252 

.86457 

.51753 

.85567 

.53238 

.84650 

.54708 

.83708 

.56160 

.82741 

50 

11 

.50277 

.86442 

.51778 

.85551 

.53263 

.84635 

.54732 

.83692 

.56184 

.82724 

49 

12 

.50302 

.86427 

.51803 

.85536 

.53288 

.84619 

.54756 

.83676 

.56208 

.82708; 

48 

13 

.50327 

.86413 

.51828 

.85521 

,53312 

.84604 

.54781 

.83660 

.56232 

.82692 

47 

14 

.50352 

.86398 

.51852 

.85506 

.53337 

.84588 

.54805 

.83645 

.56256 

.82675!  46 

15 

.50377 

.86384 

.51877 

.85491 

.53361 

.84573 

.54829 

.83629 

.56280 

. 82659 1 

45 

16 

.50403 

.86369 

.51902 

.85476 

.53386 

.84557 

.54854 

.83613 

.56305 

.82643,  44 

17 

.50428 

.86354 

.51927 

.85461 

.53411 

.84542 

.54878 

.83597 

.56329 

.82626 

43 

18 

.50453 

.86340 

.51952 

.85446 

.53435 

84526 

.54902 

.83581 

.56353 

.82610 

42 

19 

.50478 

.86325 

.51977 

.85431 

.53460 

.84511 

.54927 

.83565 

.56377 

.82593 

41 

20 

.50503 

.86310 

.52002 

.85416 

.53484 

.84495 

.54951 

.83549 

.56401 

.82577; 

40 

21 

.50528 

.86295 

.52026 

.85401 

.53509 

.84480 

.54975 

.83533 

.56425 

. 82561 1 

[ 39 

22 

. 50553 

.86281 

.52051 

.85385 

.53534 

.84464! 

54999 

.83517 

.56449 

.82544 

1 38 

23 

.50578 

.86266 

.52076 

.85370 

.53558 

.84448! 

.55024 

. 83501 

.56473 

.82528!  37 

24 

.50603 

.86251 

.52101 

.85355 

.53583 

.84433 

.55048 

-83485! 

.56497 

.82511 

36 

25 

.50628 

.86237 

.52126 

.85340 

.53607 

.844171 

.55072 

83469! 

.56521 

.82495!  35 

26 

.50654 

.86222 

.52151 

.85325 

.53632 

.84402 

.55097 

1.83453 

.56545 

.82478 

1 34 

27 

.50679 

.86207 

.52175 

.85310 

.53656 

.84386 

.55121 1 

.83437! 

.56569 

.82462 

33 

28 

.50704 

.86192 

.52200 

.85294 

.53681 

.84370 

.55145 

.83421 ! 

.56593 

.82446 

32 

29 

.50729 

.86178 

.52225 

.85279 

.53705 

.84355 

.55169 

.83405 

.56617 

.82429 

31 

30 

.50754 

.86163 

.52250 

.85264 

.53730 

.84339 

.55194 

.83389 

56641 

.82413 

30 

31 

.50779 

.86148 

.52275 

.85249 

.53754 

.84324 

.55218 

.83373 

,56665 

.82396 

29 

32 

.50804 

.86133 

.52299 

.85234 

. 53779 

.843081 

.55242 

.83356 

.56689 

82380 

28 

33 

.50829 

.86119 

.52324 

.85218 

.53804 

.84292! 

.55266 

.83340 

.56713 

.82363 

27 

34 

.50854 

.86104 

.52349 

.85203 

.53828 

.84277! 

.55291 

.83324 

.56736 

.82347 

1 26 

35 

.50879 

.86089 

.52374 

.85188 

.53853 

.84261 

.55315 

.83308 

.56760 

.82330 

! 25 

36 

.50904 

.86074 

.52399 

.85173 

.53877 

.84245! 

.55339 

.83292 

.56784 

.82314 

! 24 

37 

.50929 

.86059 

.52423 

.85157 

.53902 

.84230 

.55363, 

.83276! 

.56808 

.82297 

! 23 

38 

.50954 

.86045 

.52448 

.85142 

.53926 

,84214! 

.55388 

.83260 

.56832 

.82281 

; 22 

39 

.50979 

.86030 

.52473 

.85127 

.53951 

.84198 

.55412 

.83244! 

.56856 

.82264 

21 

40 

.51004 

86015 

.52498 

.85112 

.53975 

.84182 | 

.55436 

. 83228 j 

.56880 

GO 

-H 

00 

20 

41 

.51029 

.86000 

.52522 

.85096 

.54000 

.84167 

.55460 

.83212' 

! .56904 

.82231 

19 

42 

.51054 

.85985 

.52547 

.85081 

.54024 

.84151 

.55484 

|.83195i 

.56928 

.82214 

j 18 

43 

.51079 

.85970 

.52572 

.85066 

.54049 

.84135 

.55509 

1.83179; 

.56952 

.82198 

17 

44 

.51104 

.85956 

.52597 

.85051 

.54073 

.84120 

.55533 

!. 83163! 

i .56976 

.82181 

i 16 

45 

.51129 

.85941 

.52621 

.85035 

.54097 

.84104 

.55557 

1.83147! 

I .57000 

.82165 

15 

46 

.51154 

.85926 

.52646 

.85020 

.54122 

: 84088 

.55581 

.83131| 

i .57024 

.82148 

14 

47 

.51179 

.85911 

.52671 

.85005 

.54146 

.84072 

.55605 

.83115; 

! .57047 

.82132 

13 

! 48 

.51204 

.85896 

.52696 

.84989 

.54171 

.84057 

.55630 

.83098 

i .57071 

.82115 

12 

| 49 

.51229 

.85881 

.52720 

.84974 

.54195 

.84041 

.55654 

.83082! 

j .57095 

.82098 

11 

| 50 

.51254 

.85866 

.52745 

.84959 

.54220 

.84025 

.55678 

.83066 

■ .57119 

.82082 

10 

51 

.51279 

.85851 

.52770 

.84943 

.54244 

.84009 

.55702 

.83050' 

.57143 

.82065 

9 

52 

.51304 

.85836 

.52794 

.84928 

.54269 

.83994 

.55726 

.83034 ; 

.57167 

.82048 

8 

53 

.51329 

.85821 

.52819 

.84913 

.54293 

.83978 

.55750 

.83017, 

.57191 

.82032 

7 

54 

.51354 

.85806 

.52844 

.84897 

.54317 

.83962 

. 55775 

.83001 i 

.57215 

.82015 

6 

55 

.51379 

.85792 

.52869 

.84882 

.54342 

.83946 

.55799 

.82985| 

.57238 

.81999 

5 

56 

.51404 

.85777 

.52893 

.84866 

.54366 

.83930 

.55823 

.82969 

.57262 

.81982 

4 

57 

.51429 

.85762 

.52918 

.84851 

.54391 

.83915 

.55847 

.82953 

.57286 

.81965 

3 

58 

.51454 

.85747 

.52943 

.84836 

.54415 

.83899 

.55871 

.82936 

.57310 

.81949 

2 

59 

.51479 

.85732 

.52967 

.84820 

.54440 

1.83883 

.55895 

.82920' 

.57334 

.81932 

1 

60 

.51504 

.85717 

.52992 

.84805 

.54464 

1.83867 

.55919 

.82904 

.57358 

.81915 

0 

/ 

Cosin 

Sine 

Cosin 

Sine 

Cosin  | Sine  j 

Cosin 

Sine 

Cosin 

Sine 

! f 

59° 

58° 

57°  i 

o 

CO 

lO 

55“  1 

TABLE  I.  SINES  AND  COSINES. 


13? 


35° 

o 

CO 

CO 

o 

00 

o 

OO 

CO 

39° 

Sine 

Cosin 

Sine 

Cosin  | 

Sine 

Cosin 

Sine 

Cosin 

Sine  I 

Cosin 

-o 

.57358 

.81915 

.58779 

.809021 

.60182 

.79864 

.61566 

T78801 

762932 

.77715 

60 

1 

.57381 

.81899 

.58802 

.808851 

.60205 

.79846 

.61589 

.78783 

.62955 

.77696 

59 

2 

.57405 

.81882 

.58826 

.80867! 

.60228 

.79829 

.61612 

.78765 

.62977, 

.77678 

58  | 

3 

.57429 

.81865 

.58849 

.80850! 

.60251 

.79811 

.61635 

.78747 

.63000' 

.77660 

57 

4 

.57453 

.81848! 

.58873 

.80833! 

.60274 

.79793 

.61658 

.78729 

. 63022 1 

.77641 

56  ! 

5 

.57477 

.81832 

.58896 

.80816 

.60298 

.79776 

.61681 

.78711 

.63045 

.77623 

55  ] 

6 

.57501 

.81815 

.58920 

.80799] 

.60321 

.79758 

.61704 

.78694 

.63068 

.77605 

54 

7 

.57524 

.81798 

.58943 

.80782! 

.60344 

.79741 

.61726 

.78676 

.63090 

.77586 

53 

8 

.57548 

.81782 

.58967 

.80705! 

.60367 

.79723 

.61749 

.78658 

.63113 

.77568 

ro 

9 

.57572 

.81765 

.58990 

. 807’48 ! 

.60390 

.79706 

.61772 

.78640 

.63135 

.77550 

51 

10 

.57596 

.81748 

.59014 

.80730 

.60414 

.79688 

.61795 

.78622! 

.63158 

.77531 

50 

11 

.57619 

.81731 

.59037 

.80713 

.60437 

.79671 

.61818 

.78604 

.63180 

.77513 

40 

12 

.57643 

.81714 

.59061 

.80696 

.60460 

.79653 

.61841 

.78586 

. C3203 1 

.77494 

48  , 

13 

.57667 

.81698 

.59084 

.80679 

.60483 

.79635 

.01864 

.78568 

.63225 

.77476 

47  ] 

14 

.57691 

.81681 

.59108 

Si 

.C050G 

.79618 

.61887 

.78550; 

.63248 

.77458 

46 

15 

.57715 

.81664 

.59131 

.60529 

.79600 

.61909 

.78532 

.63271. 

.77439 

43  j 

16 

.57738 

.81647 

.59154 

.80627! 

.60553 

.79583 

.61932 

.78514 

.63293 

.77421 

44 

17 

.57762 

.81631 

.59178 

.80610’ 

.60576 

.79565 

.61955 

.78496 

.63316 

.77402 

43  1 

18 

.57786 

.81614 

.59201 

.80593 

.60599 

.79547 

.61978 

.78478 

.63338 

. 77384 

42 

19 

.57810 

.81597 

.59225 

.80576,, 

. 60622 

.79530 

.62001 

1.78460, 

.63361 

.77366 

41 

20 

.57833 

.81580 

.59248 

.80558; 

.60645 

.79512 

.62024 

.78442 

.63383, 

.77347 

40 

21 

.57857 

.815631 

.59272 

.80541 

.60668 

.79494 

.62046 

.78424 

.63406! 

.77329 

39 

22 

.57881 

.81546 

.59295 

.80524 

.60691 

.79477 

! .62069 

. 78405 1 

.63428] 

.77310 

88 

23 

.57904 

.815301 

.59318 

. 80507 i 

.60714 

.79459 

.62092 

.78387! 

.63451 

.63473 

77292 

37 

24 

.57928 

.81513 

.59342 

.80489 

.60738 

.79441 

.62115 

.78369] 

! 77273 

86 

25 

.57952 

.81496! 

.59365 

.80472 

.60761 

.79424 

.62138 

.78351 1 

. 63496 

.77255 

35 

26 

.57976 

.81479! 

.59389 

.80455 

.60784 

.79406 

.62160 

! .78333 

.63518 

.77236 

34 

27 

.57999 

.81462! 

.59412 

.80438 

.60807 

.79388 

: .62183 

; .7  8315 

.63540 

.77218 

33 

28 

.58023 

.81445! 

.59436 

.80420: 

.60830 

.79371 

j .62206 

i .78297 

.63563 

.77199 

32 

29 

.58047 

.8142# 

.59459 

.80403] 

60853 

.79353 

62229 

.78279 

.63585 

.77181 

31 

30 

.58070 

.81412 

.59482 

.80386, 

.60876 

.79335 

.62251 

.78261 

.63608 

.77162 

30 

31 

.58094 

.81395 

..59506 

. 80368 ! 

.60899 

.79318 

.62274 

.78243 

.63630 

. 77144 

29  ! 

32 

.58118 

.81378 

.59529 

.80351 | 

.60922 

.79300 

.62297 

.78225 

.63653 

.77125 

28  | 

33 

34 

.58141 

.58165 

.813611 

.81344] 

.59552 

.59576 

.80334! 
.80316 ! 

.60945 

.60968 

.79282 

.79264 

.62320 
! .62342 

.78206 

.78188 

.63675 

.63698 

.77107 

.77088 

27 

26 

35 

.58189] 

.81327! 

.59599 

.80299] 

.60991 

.79247 

i .62365 

j. 78170 

.63720 

.77070!  25 

36 

.582121 

.81310] 

.59622 

.80282! 

.61015 

.79229 

.62388 

.78152| 

.63742 

.77051 

! 24  .j 

37 

.58236] 

.81293 

. 59646 - 

*.80264  i 

.61038 

.79211 

.62411 

.78134 

.63765 

1.77033 

23 

38 

.58260 

.812761 

.59669 

.80247] 

.61061 

.79193 

.62433 

.78116! 

.78098! 

.63787 

.77014!  22  i 

39 

.58283 

.81259 

.59693 

.80230! 

.61084 

.79176 

.62456 

.63810 

. 76996 

21 

40 

.58307 

1.81242 

.59716 

.80212 | 

.61107 

.79158 

.62479 

.78079] 

.63832 

!■ 76977 

i 20  I 

41 

.58330 

. 81225 1 
] .81208 ! 

.59739 

.80195 

.61130 

.79140 

.62502 

.78061 ! 

.63854 

!. 76959:  19 

42 

.58354 

.59763 

.80178] 

.61153 

.79122 

.62524 

.78043 

.63877 

.76940 

! 18 

43 

.58378 

.81191! 

.59786 

.80160 

.61176 

.79105 

.62547 

.78025 

.63899 

.76921 

17 

44 

.58401 

! .81174 

.59809 

.80143 

.61199 

.79087 

.62570 

.78007 

.63922 

.76903 

! 16 

45' 

.58425 

i .81157 

.59832 

.80125 

.61222 

.79069 

.62592 

.77988 

.63944 

. 7 6884 

15 

46 

.58449 

] .8*1140! 

. 59856 

.80108 

.61245 

.79051 

.62615 

.77970 

.63966 

.76866 

14 

47 

.58472 

.81123| 

.59879 

.80091 

.61268 

.79033 

.62638 

.77952 

.63989 

.76847 

13 

48 

.58496 

.81106 

.59902 

.80073 

.61291 

.79016 

.62660 

.77934 

.64011 

. 76828 

12 

49 

.58519 

.81089 

.59926 

.80056] 

.61314 

.78998 

! .62683 

.77916 

.64033 

i. 76810 

11 

50 

.58543 

.81072 

.59949 

.80038; 

.61337 

.78980 

.62706 

.77897, 

.64056 

.76791 

19 

51 

. 58567 

.81055 

.59972 

.80021 

.61360 

.78962 

.62728 

.77879] 

.64078 

.76772 

9 

52 

.58590 

.81038 

.59995 

80003 

.61383 

.78944! 

! .62751 

1.77861 

.64100 

.76754 

! 8 ! 

53 

.58614 

.81021! 

.60019 

.79986 

.61406 

.78926! 

.62774 

00 

1 

.64123 

.76735 

j 7 

54 

.58637 

.81004 

.60042 

.79968 

.61429 

.78908 

.62796 

.77824 

.64145 

.70717 

! 6 ’ 

55 

.58661 

.80987 

.60065 

.79951 

.61451 

.78891! 

.62819 

.77806: 

.64167 

j. 7 6698 

! 5 

56 

.58684 

.80970 

.60089 

.79934 

.61474 

.78873 

.62842 

.77788! 

.64190 

1.76679 

4 

57 

.58708 

.80953 

.60112 

.79916 

.61497 

.78855 

.62864 

.77769 

.64212 

.76661 

3 

58 

.58731 

.80936 

.60135 

.79899 

.61520 

.78837 

.62887 

.77751 

.64234 

.76642 

2 

59 

.58755 

.80919 

.60158 

.79881 

.61543 

.78819 

.62909 

.77733 

.64256 

.76623 

i 1 

60 

.58779 

.80902 

.60182 

.79864 

.61566 

.78801 

.62932 

.77715 

.64279 

.76604 

j 0 

9 

Cosin 

Sine 

Cosin 

Sine  j 

Cosin 

Sine 

! Cosin 

Sine 

Cosin  | 

Sine 

/ 

1 54° 

53° 

52° 

51° 

50° 

138 


TABLE  I.  SINES  AND  COSINES. 


40° 

41° 

42° 

0 

0 

Sine 

Cosin 

Sine 

Cosin 

Sine 

Cosin 

Sine 

Cosin 

Sine 

Cosin 

f 

0 

.64279 

.76604 

. 65606 

.75471 

.66913 

.74314 

768200 

.73135 

.69466 

.71934 

60 

1 

.64301 

.76586 

.65628 

.75452 

.66935 

.74295 

.68221 

.73116 

.69487 

.71914 

59 

2 

.64323 

.76567 

.65650 

.75433 

.66956 

.74276 

.68242 

.73096 

.69508 

.71894 

58 

3 

.64346 

.76548 

.65672 

.75414 

.66978 

.74256 

.68264 

.73076 

.69529 

.71873 

57 

4 

.64368 

.76530 

.65694 

.75395 

.66999 

.74237 

.68285 

.73056 

.69549 

.71853 

56 

5 

.64390 

.76511 

.65716 

. 75375 

.67021 

.74217 

.68306 

.73036 

.695?‘0 

.71833 

55 

6 

.64412 

.76492 

.65738 

.75356 

.67043 

.74198 

.68327 

.73016 

.69591 

.71813 

54 

7 

.64435 

.76473 

.65759 

.75337 

.67064 

.74178 

.68349 

.72996 

.69612 

.71792 

53 

8 

.64457 

.76455 

.65781 

.75318 

.6r086 

.74159 

.68370 

.72976 

.69633 

.71772 

53 

9 

.64479 

.76438 

.65803 

.75299 

.67107 

.74139 

.68391 

.72957 

.69654 

.71752 

51 

10 

.64501 

.76417 

.65825 

.75280 

.67129 

.74120 

.68412 

.72937 

.69675 

.71732 

50 

It 

.64524 

.76398 

.65847 

.75261 

.67151 

.74100 

.68434 

.72917 

.69696 

.71711 

49 

12 

.64546 

.76380 

. 65889 

.75241 

.67172 

.74080 

.68455 

.72897 

.69717 

.71691 

48 

13 

.64568 

.76381 

.65391 

.75222 

.67194 

.74061 

.68476 

.72877 

.69737 

.71671 

47 

14 

.64590 

78342 

.65913 

.75203 

.67215 

.74041 

.68497 

.72857 

.69758 

.71650 

i 46 

15 

.64612 

.76323 

.65935 

.75184 

.67237 

.74022 

.68518 

.72837 

.69779 

.71630 

i 45 

16 

.64635 

.76304 

. 65956 

.75165 

.67258 

.74002 

.68539 

.72817 

.69800 

.71610 

44 

17 

.64657 

.76286 

.65978 

.75146 

.67280 

. 7d983 

.68561 

.72797 

.69821 

.71590 

! 43 

! 18 

.64679 

.76267! 

.66000 

.75128 

.67301 

.73963 

.68582 

.72777 

.69842 

.71569 

42 

19 

.64701 

.76248! 

.68022 

.75107] 

.67323 

.73944 

.68603 

.72757 

.69862 

.71549 

41 

20 

.64723 

.70229: 

.66044 

.75088 

.67344 

.73924 

.68624 

. 7273?' 

.69883 

.71529 

40 

21 

.64746 

.76210' 

.66066 

.75089 

.67366 

.73904 

.68645 

.72717 

.69904 

.71508 

39 

22 

.64768 

.76192 

.68038 

.75050 

.67387 

.73885 

.68666 

.72697 

.69925 

.71488 

38 

23 

.64790 

.76173! 

.68109 

.75030 

.67409 

. 73865 

.68688 

.72677; 

.69946 

.71468 

37 

24 

.64812 

.76154 

.68131 

.75011 

.67430 

.73846 

.68709 

,72857 

.69966 

.71447 

36 

25 

.64834 

.76135 

.68153 

.749921 

.67452 

.73826 

.68730 

.72637' 

.69987 

.71427 

35 

26 

.64856 

.76116 

.63175 

.74973 

.67473 

.73806 

.68751 

.72617 ! 

.70008 

.71407 

34 

27 

.64878 

.76097j 

.66197 

.74953 

.67495 

.73787 

.68772 

.72597! 

.70029 

.71386 

33 

28 

.64901 

.76078' 

.68218 

.74934 

.67516 

.73767 

.68793 

.725771 

.70049 

.71366 

32 

29 

.64923 

.76059 

.66240 

.74915 

.67538 

.73747 

.68814 

.72557' 

.70070 

.71345 

31 

30 

.64945 

.76041 

.66262 

.74896 

.67559 

.73728 

.68835 

.72537 

.70091 

.71325|  30 

31 

.64967 

.76022 

.68284 

.74876 

.67580 

.73708 

.68857 

.72517 

.70112 

.713051  29 

32 

.64989 

.76003 

.66306 

.74857 

.67602 

.73688 

.68878 

.72497 

70132 

.71284!  28 

33 

.65011 

.75984 

.66327 

.74838 

.67623 

.73669 

.68899 

. 72477 

.70153 

.71264  27 

34 

.65033 

.75965 

.68349 

.74818 

.67645 

.73649 

.68920 

.72457 

.70174 

.71243!  26 

35 

.65055 

.75946' 

.68371 

.74799 

. 67666 

.73629 

.68941 

.72437 

.70195 

.71223 

35 

36 

.65077 

.75927! 

.66393 

.74780 

.67688 

.73610 

.68962 

.7241? 

.70215 

.712031 

24 

37 

.65100 

.75908 

.68414 

.74760; 

.67709 

.73590 

.68983 

.72397 

.70236 

.71182'  23 

38 

.65122 

.75889 

.66436 

.74741 

.67730 

73570 

.69004 

. 72377 

.70257 

.71162;  22 

39 

.65144 

.75870 

.66458 

.74722 ! 

.67752 

.73551 

.69025 

.72357 

.70277 

.711411 

21 

40 

.65166 

.75851! 

.66480 

.74703  | 

.67773 

.73531 

.69046 

.72337 

.70298 

.71121! 

20 

41 

.65188 

-.75832; 

.66501 

.74683 

.67795 

.73511 

.69067 

.72317 

.70319 

.711001 

19 

42 

.65210 

.75813; 

.68523 

.74664' 

.67816 

.73491 

.69088 

.72297 

.70339 

.71080 

18 

43 

.65232 

.75794 

. 68545 

.74644| 

.67837 

.73472 

.69109 

72277 

.70360 

.71059 

17 

44 

. 65254 

.75775 

. 66566 

.74625; 

.67859 

.73452 

.69130 

! 72257 

.70381 

.71039  16 

45  ! 

.65276 

.75756; 

.66588 

.74606j 

.67880 

.73432 

.69151 

.72236 

.70401 

.71019'  15 

46  j 

.65298 

.75738 

.66610 

.74586  | 

.67901 

.73413 

.69172 

.72216 

.70422 

.70998! 

14 

47  ! 

.65320 

.75719: 

.66632 

.745671 

.67923 

.73393 

.69193 

.72196 

.70443 

.70978  13 

48 

.65342 

.757001 

.66653 

.74548' 

.67944 

.73373 

.69214 

.72176 

.70463 

.70957 

12 

49 

.65364 

.75680; 

.66675 

.74528 

.67965 

.73353 

.69235 

.72156 

.70484: 

.70937 

11 

50  , 

.65386 

.75661 

.66697 

.74509 

.67987 

.73333 

.69256 

.72136 

.70505 

.70916j  10 

51  ' 

.65408 

.75642 

.66718 

.74489! 

.68008 

.73314 

.69277 

.72116 

.70525 

.70896! 

9 

52 

.65430 

.75623 

.66740 

.74470; 

.68029 

.73294 

.69298 

.72095 

. ?0546 

.70875 

8 

53 

.65452 

.75604 | 

.66762 

.74451 ! 

.68051 

.73274 

.69319 

.72075 

.70567! 

.70855; 

7 

54 

.65474 

.755851 

.66783 

.74431 

.68072 

.73254 

.69340 

.72055 

.70587! 

.70834: 

6 

55 

. 65496 

. 75566 ! 

.66805 

.74412 

.68093 

.73234 

.69361 

.72035 

.70608' 

.70813 

5 

56 

.65518 

. 75547 1 

.66827 

.74392 

.68115 

.73215 

.69382 

.72015 

.70628! 

.70793 

4 

57 

.65540 

.75528! 

.66848 

.74373 

.68136 

.73195 

.69403 

.71995 

.70649! 

.70772 

3 

58 

.65562 

.75509; 

.66870! 

.74353 

.68157 

.73175 

.69424 

.719?4 

.70670 

.70752 

2 

59 

.65584 

.75490 

.66891 

.74334 

.68179 

.73155 

.69445 

.71954 

.70690! 

.70731; 

1 

60 

.65606 

.75471 ! 

.66913 

.74314 

.68200 

.73135 

.69466 

.71931' 

.70711 ; 

.70711 

0 

/ 

Cosin 

Sine 

Cosin 

Sine  ! 

Cosin 

Sine 

Cosin 

Sine  | 

Cosin 

Sine  j 

/ 

49° 

co 

0 

1 47° 

0 

CO 

45°  1 

Table  II. 


NATURAL  TANGENTS  AND  COTANGENTS 

TO 


FIVE  DECIMAL  PLACES. 


140  TABLE  II.  TANGENTS  AND  COTANGENTS. 


0° 

L°  ! 

2°  |l  3° 

Tang 

Cotang 

Tang 

Cotang 

Tang 

Cotang 

| Tang 

Cotang 

[ / 

0 

.00000 

Infinite. 

.01746 

57.2900 

.03492 

28.6363 

.05241 

19.0811 

60 

1 

.00029 

3437.75 

.01775 

56.3506 

.03521 

28.3994 

.05270 

18.9755 

59 

2 

.00058 

1718.87 

.01804 

55.4415 

.03550 

28.1604 

.05299 

18.8711 

158 

3 

.00087 

1145.92 

.01833 

54.5613 

.03579 

27.9372 

.05328 

18.7678 

57 

4 

.00116 

859.436 

.01862 

53.7036 

.03609 

27.7117 

.05357 

18.6656 

56 

5 

.00145 

687.549 

.01891 

52.8821 

.03638 

27.4899 

.05387 

18.5045 

55 

6 

.00175 

512.957 

.01920 

52.0807 

.03667 

27.2715 

.05416 

18.4645 

54 

7 

.00204 

491.106 

.01949 

51.3032 

.03696 

27.0506 

.05445 

18.3655 

53 

8 

.00233 

429.718 

.01978 

50.5485 

.03725 

26.8450 

.05474 

18.2677 

52 

9 

.00262 

381.971 

.02007 

40.8157 

.03754 

26.6367 

.05503 

18.1708 

51 

10 

.00291 

343.774 

.02036 

49.1039 

.03783 

26.4316 

.05533 

18.0750 

56 

11 

.00320 

312  521 

.02066 

48.4121 

.03812 

26.2296 

.05562 

17.9802 

49 

12 

.00349 

286.478 

.02095 

47.7395 

.03842 

26.0307 

.05591 

17.8863 

48 

13 

.00378 

264.441 

.02124 

47.0853 

.03871 

25.8348 

.05020 

17.7934 

47 

14 

.00407 

245.552 

.02153 

46.4489 

.03900 

25.6418 

.05649 

17.7015 

46 

15 

.00436 

229.182 

.02182 

45.8294 

.03929 

25.4517 

.05078 

17.6106 

45 

16 

.00465 

214.858 

.02211 

45.2261 

.03958 

25.2044 

.05708 

17.5205 

44 

17 

.00495 

202.219 

.02240 

44.6386 

.03987 

25.0798 

.05737 

17.4314 

43 

18 

.00524 

190.984 

.02269 

44.0661 

.04016 

24.8978 

.05766 

17.3432 

42 

19 

.00553 

180.932 

.02298 

43.5081 

.04046 

24.7185 

.05795 

17.2558 

41 

20 

.00582 

171.885 

.02328 

42.9641 

.04075 

24.5418 

.05824 

17.1693 

40 

21 

.00611 

163,700 

.02357 

42.4335 

.04104 

24.3675  ' 

.05854 

17.0837 

39 

22 

.00640 

156.259 

.02386 

41.9158 

.04133 

24.1957 

.05883 

16.9990 

38 

23 

.00669 

149.465 

.02415 

41.4106 

.04162 

24.0263 

.05912 

16.9150 

37 

24 

.00698 

143.237 

.02444 

40.9174 

.04191 

23.8593 

.05941 

16.8319 

36 

25 

.00727 

137.507 

.02473 

40.4358 

.04220 

23.6945 

. 05970 

16.7496 

35 

26 

’.00756 

132.219 

.02502 

39.9655 

.04250 

23.5821 

.05999 

16.6681 

34 

27 

.00785 

127.321 

.02531 

39.5059 

.04279 

23.3718 

.00029 

16.5874 

33 

28 

.00815 

122.774 

.02560 

39.0568 

.04308 

23.2137 

.06058 

16.5075 

32 

29 

.00844 

118.540 

.02589 

38.6177 

.04337 

23.0577 

.06087 

16.4283 

31 

30 

.00873 

114.589 

.02619 

38.1885 

.04366 

22.9038 

.06116 

16.3499 

30 

31 

.00902 

110.892 

.02648 

37.7680 

.04395 

22.7519 

.06145 

16.2722 

29 

32 

.00931 

107.426 

.02677 

37.3579 

.04424 

22.6020 

.06175 

16.1952 

28 

33 

.00960 

104.171 

.02706 

36.9560 

.04454 

22.4541 

.06204 

16.1190 

27 

34 

.00989 

101.107 

.02735 

36.5627 

.04483 

22.3081 

.06233 

16.0435 

26 

35 

.01018 

98.2179 

.02764 

36.1776 

.04512 

22.1640 

.06262 

15.9687 

25 

36 

.01047 

95.4895 

.02793 

35.8006 

.04541 

22.0217 

.06291 

15.8945 

24 

37 

.01076 

92.9085 

.02822 

35.4313 

.04570 

21.8813 

.06321 

15.8211 

23 

38 

.01105 

90.4663 

.02851 

35.0695 

.04599 

21.7426 

.06350 

15.7483 

22 

39 

.01135 

88.1436 

.02881 

34.7151 

.04628 

21.6056 

.06379 

15.6762 

21 

40 

.01164 

85.9398 

.02910 

34.3678 

.04658 

21.4704 

.06408 

15.6048 

20 

41 

.01193 

83.8435 

.02939 

34.0273 

.04687 

21.3369 

.06437 

15.5340 

19 

42 

.01222 

81.8470 

.02968 

33.6935 

.04716 

21.2049 

.06467 

15.4638 

18 

43 

.01251 

79.9434 

.02997 

33.3662 

.04745 

21.0747 

.06496 

15.3943 

17 

44 

.01280 

78.1263 

.03026 

33.0452 

.04774 

20  9460 

.06525 

15.3254 

16 

45 

.01309 

76.3900 

.03055 

32.7303 

.04803 

20.8188 

.06554 

15.2571 

15 

46 

.Cl 338 

74.7292 

.03084 

32.4213 

.04833 

20.6932 

.06584 

. 15.1893 

14 

47 

.01367 

73.1390 

.03114 

32.1181 

.04862 

20.5691 

.06613 

15.1222 

13 

48 

.01396 

71.6151 

.03143 

31.8205 

.04891 

20.4465 

.06642 

15.0557 

12 

49 

.01425 

70.1533 

.03172 

31.5284 

.04920 

20.3253 

.06671 

14.9898 

11 

50 

.01455 

68.7501 

.03201 

31.2416 

.04949 

20.2056 

.06700 

14.9244 

10 

51 

.01484 

67.4019 

.03230 

30.9599 

.04978 

20.0872 

.06730 

14.8596 

9 

52 

.01513 

66.1055 

.03259 

30.6833 

.05007 

19.9702 

.06759 

14.7954 

8 

53 

.01542 

64.8580 

.03288 

30.4116 

.05037 

19.8546 

.06788 

14.7317 

7 

54 

.01571 

63.6567 

.03317 

30.1446 

.05066 

19.7403 

.06817 

14.6685 

6 

55 

.01600 

62.4992 

.03346 

29.8823 

.05095 

19.6273 

.06847 

14.6059 

5 

56 

.01629 

61.3829 

.03376 

29.6245 

.05124 

19.5156 

.06876 

14.5438 

4 < 

57 

.01658 

60.3058 

.03405 

29.3711 

.05153 

19.4051 

.06905 

14.4823 

3 

58 

.01687 

59  2659 

.03434 

29.1220 

.05182 

19.2959 

.06934 

14.4212 

2 

59 

.01716 

58.2612 

.03463 

28.8771 

.05212 

19.1879 

.06963 

14.3607 

1 

60 

.01746 

57.2900 

.03492 

28.6363 

.05241 

19.0811 

.06993 

14.3007 

0 

f 

Cotang 

Tang 

Cotang 

Tang 

Cotang 

Tang 

Cotang  | 

Tang 

f I 

89° 

88°  1 

87° 

o 

CD 

QO 

1 

TABLE  IT, 


TANGENTS  AND  COTANGENTS. 


141 


4 

o | 

£ 

i* 

€ 

7 

Tang 

Cotang 

Tang 

Cotang 

Tang  1 Cotang  ! 

Tang  | 

Cotang 

0 

.06993 

14.3007 

.08749 

11.4301 

.10510 

9.51436 

.12278 

8.14435 

60 

1 

.07022 

14.2411  ! 

.08778 

11.3919 

.10540 

9.48781 

.12308 

8.12481 

59 

2 

.07051 

14.1821  1 

.08807 

11.3540 

i . 10569  . 

•9.46141 

.12338 

8.10536 

58 

3 

.07080 

14.1235 

.08837 

11.3163 

.10599 

9.43515 

.12367 

8.08600 

57 

4 

.07110 

14.0655  I 

.08866 

11.2789 

| .10628 

9.40904 

.12397 

8.06674 

56 

5 

.07139 

14.0079 

.08895 

11.2417 

.10657 

9.38307 

.12426 

8.04756 

55 

G 

.07168 

13.9507 

.08925 

11.2048 

.10687 

9.35724 

.12456 

8.02848 

54 

7 

.07197 

13.8940 

.08954 

11.1681 

.10716 

9.33155 

.12485 

8.00948 

53 

8 

.07227 

13.8378 

.08983 

11.1316 

. 107'46 

9.30599 

.12515 

7.99058 

52  i 

9 

.07256 

13.7821 

.09013 

11.0954 

.10775 

9.28058 

.12544 

7.97176 

51  ; 

10 

.07285 

13.7267 

.09042 

11.0594 

.10805 

9.25530 

.12574 

7.95302 

50  f 

11 

.07314 

13.6719 

.09071 

11.0237 

.10834 

9.23016 

.12603 

7.93438 

49 

12 

.07344 

13.6174 

.09101 

10.9882 

.10863 

9.20516 

.12633 

7.91582 

48 

13 

.07373 

13.5634 

.09130 

10.9529 

.10893 

9.18028 

.12662 

7.89734 

47  ■ 

14 

.07402 

13.5098 

.09159 

10.9178 

.10922 

9.15554 

.12692 

7.87895 

46 

15 

.07431 

13.4566 

.09189 

10.8829- 

.10952 

9.13093 

.12722 

7.86064 

45 

1G 

.07461 

13.4039 

.09218 

10.8483 

.10981 

9.10646 

.12751 

7.84242 

44 

17 

.07490 

13.3515 

.09247 

10.8139 

.11011 

9.08211 

.12781 

7. 82428% 

43 

18 

.07519 

13.2996 

.09277 

10.7797 

.11040 

9 . 057'89 

.12810 

7.80622 

42 

19 

.07548 

13.2480 

.09306 

10.7457 

.11070 

9.03379 

.12840 

7.78825 

41 

20 

.07578 

13.1969 

.09335 

10.7119 

.11099 

9.00983 

.12869 

7.77035 

40 

21 

.07607 

13.1461 

'.09365 

10.6783 

.11128 

8.98598 

.12899 

7.75254 

39 

22 

.07636 

13.0958 

.09394 

10.6450 

.11158 

8.96227 

.12929 

7.73480 

38 

23 

. 07665 

13.0458 

.09423 

10.6118 

.11187 

8.93867 

.12958 

7.71715 

37 

24 

.07695 

12.9962 

.09453 

10.5789 

.11217 

8.91520 

.12988 

7.69957 

36 

25 

.07724 

12.9469 

.09482 

10.5462 

.11246 

8.89185 

.13017 

7.68208 

35 

26 

.07753 

12.8981 

.09511 

10.5136 

.11276 

8.86862 

.13047 

7.66466 

34 

27 

.07782 

12.8496 

.09541 

10.4813 

.11305 

8.84551 

.13076 

7.64732 

33 

28 

.07812 

12.8014 

.09570 

10.4491 

.11335 

8.82252 

.13106 

7.63005 

32 

29 

.07841 

12.7536 

.09600 

10.4172 

.11364 

8.79964 

.13136 

7.61287 

31 

30 

.07870 

12.7062 

.09629 

10.3854 

.11394 

8.77689 

.13165 

7.59575 

30 

31 

.07899 

12.6591 

.09658 

10.3538 

.11423 

8.75425 

.13195 

7.57872 

29 

32 

.07929 

12.6124 

.09688 

10.3224 

.11452 

8.73172 

.13224 

7.56176 

28 

33 

.07958 

12.5G60 

.09717 

10.2913 

.11482 

8.70931 

.13254 

7.54487 

27 

34 

.07987 

12.5199 

.09746 

10.2602 

.11511 

8.68701 

.13284 

7.52806 

26 

35 

.08017 

12.4742 

.09776 

10.2294 

.11541 

8.66482 

.13313 

7.51132 

25 

36 

.08046 

12.4288 

.09805 

10.1988 

.11570 

8.64275 

.13343 

7.49465 

24 

37 

.08075 

12.3838 

.09834 

10.1683 

1 .11600 

8.62078 

.13372 

7.47806 

23 

38 

.08104 

12.3390 

.09864 

10.1381 

! .11629 

8.59893 

.13402 

7.46154 

22 

39 

.08134 

12.2946 

.09893 

10.1080 

.Ilf  » 

8.57718 

.13432 

7.44509 

21 

40 

.08163 

12.2505 

.09923 

10.0780 

.11688 

8.55555 

.13461 

7.42871 

20 

41 

.08192 

12.2067 

.09952 

10.0483 

.11718 

f. 53402 

.13491 

7.41240 

19 

42 

.08221 

12.1632 

.09981 

10.0187 

: .11747 

8.51259 

.13521 

7.39616 

18 

43 

.08251 

12.1201 

.10011 

9.98931 

.11777 

8.49128 

.13550 

7.37999 

j 17 

44 

.08280 

12.0772 

.10040 

9.96007 

.11806 

8.47007 

.13580 

7.36389 

16 

45 

.08309 

12,0346 

.10069 

9.93101 

.11836 

8.44896 

.13609 

7.34786 

15 

46 

.08339 

11.9923 

.10099 

9.90211 

.11865 

8.42795 

.13639 

7.33190 

111 

47 

.08368 

11.9504 

.10128 

9.87338 

.11895 

8.40705 

.13669 

7.31600 

|13 

48 

.08397 

11.9087 

.10158 

9.84482 

.11924 

8.38625 

.13698 

7.30018 

1 12 

49 

.08427 

11.8673 

.10187 

9.81641 

.11954 

8.36555 

.13728 

7.28442 

11 

50 

.08456 

11.8262 

.10216 

9.78817 

.11983 

8.34496 

.13758 

7.2687'3 

10 

51 

.08485 

11.7853 

.10246 

8.76009 

712013 

8.32446 

.13787 

7.25310 

9 

52 

.08514 

11.7448 

.10275 

9.73217 

.12042 

8.30406 

.13817 

7.23754 

8 

53 

.08544 

11.7045 

.10305 

9.70441 

.12072 

8.28376 

.13846 

7.22204 

7 

54 

.08573 

11.6645 

.10334 

9.67680 

.12101 

8.26355 

.13876 

7.20661 

6 

55 

.08602 

11.6248 

.10363 

9.64935 

.12131 

8.24345 

.13906 

7.19125 

5 

56 

.08632 

11.5853 

.10393 

9.62205 

.12160 

8.22344 

.13935 

7.17594 

4 

57 

.08661 

11.5461 

. 10422 

9.59490 

.12190 

8.20352 

.13965 

7.16071 

3 

58 

.08690 

11  5072 

. 10452 

9.56791  ‘ 

.12219 

8.18370 

.13995 

7.14553 

2 

59 

.08720 

11.4685 

. 10481 

9.54106 

.12249 

8.16398 

.14024 

7.13042 

1 

60 

.08749 

11.4301 

1 .10510 

9.51436 

.12278 

8.14435 

.14054 

7.11537 

0 

/ 

Cotang 

Tang 

Cotang 

Tang 

Cotang 

Tang 

Cotang 

Tang 

/ 

o 

QO 

84° 

o 

CO 

00 

82° 

142 


TABLE  II. 


TANGENTS  AND  COTANGENTS, 


8° 

If  9® 

II  io° 

11° 

Tang 

Cotamg 

1 Tang 

Cotang 

Tang 

Cotang 

Tang 

Cotang 

0 

.14054 

7.11537 

.15838 

6,31375 

.17633 

5.67128 

.19438 

5.14455 

60 

1 

.14084 

7.10038 

.15868 

6.30189 

.17663 

5.66165 

.19468 

5.13658 

59 

2 

.11113 

7.08546 

.15898 

6.29007 

.17693 

5.65205 

.19498 

5.12862 

58 

3 

.14143 

7.07059 

.15928 

6.27829 

.17723 

5.64248 

.19529 

5.12069 

57 

4 

.14173 

7.05579 

.15958 

6.26655 

.17753 

5.63295 

.19559 

5.11279 

56 

5 

.14202 

7.04105 

.15988 

6.25486 

.17783 

5.G2344 

.19589 

5.10490 

55 

6 

.14232 

7.02637 

.16017 

6.24321 

.17813 

5.61397 

.19619 

5.09704 

54 

7 

. 14262 

6.91174 

.16047 

6. 231 CO 

.17843 

5.60452 

.19649 

5.08921 

53 

8 

.14291 

6.99718 

.16077 

G. 22003 

.17873 

5.59511 

.19680 

5.08139 

52 

9 

.14321 

6.98268 

.16107 

6.20851 

.17903 

5.58573 

.19710 

5.07360 

51 

10 

.14351 

0.96823 

.16137 

6.19703 

.17933 

5.57638 

.19740 

5.06584 

50 

11 

.14381 

G. 95385 

.16167 

6.18559 

.17963 

5.56706 

.19770 

5.05809 

49 

12 

.14410 

6.93952 

.16196 

6.17419 

.17993 

5 . 55777 

.19801 

5.05037 

48 

13 

.14440 

6.92525 

.16226 

6.16283 

.18023 

5.54851 

.19831 

5.042G7 

47 

14 

.14470 

6.91104 

.16256 

6.15151 

.18053 

5.53927 

.19861 

5.03499 

46 

15 

. 14499 

6.89688 

.16286 

6.14023  1 

.18083 

5.53007 

.19891 

5.02734 

45 

IS 

.14529 

6.88278 

.16316 

6.12899 

.18113 

5.52090 

.19921 

5.01971 

44 

17 

.14559 

6.86874 

.16346 

6.11779  I 

.18143 

5.51176 

.19952 

5.01210 

43 

18 

.14588 

6 . 85475 

.16376 

6.10664 

.18173 

5.50264 

.19982 

5.00451 

42 

19 

.14618 

0.84082 

.16405 

6.09552  1 

.18203 

5.49356 

.20012 

4.99695 

41 

20 

.14648 

6.82694 

.16435 

6.08444 

.18233 

5.48451 

.20042 

4.98940 

40 

21 

.14678 

6.81312 

.16465 

6.07340  ! 

.18263 

5.47548 

.20073 

4.98188 

39 

22 

.14707 

6.79936 

.16495 

6.06240 

• .18293 

5.46648 

.20103 

4.97438 

38 

23 

.14737 

6.78564 

.16525 

6.05143  ! 

.18323 

5.45751 

.20133 

4.96690 

37 

24 

.14767 

6.77199 

. 16555 

6.04051 

.18353 

5.44857 

.20164 

4 . 95945 

36 

25 

.14796 

6.75838 

.16585 

6.02962 

.18384 

5.43966 

.20194 

4.95201 

35 

26 

.14826 

6.74483 

.16615 

6.01878 

.18414 

5.43077 

.20224 

4.94460 

34 

27 

. 14856 

6.73133 

.16645 

6.00797 

.18444 

5.42192 

.20254 

4.93721 

33 

28 

. 14886 

6.71789 

.16674 

5.99720 

.18474 

5.41309 

'.20285 

4.92984 

32 

29 

.14915 

6.70450 

.16704 

5.98646 

.18504 

5.40429 

.20315 

4.92249 

31 

30 

.14945 

6.69116 

.16734 

5.97576 

.18534 

5.39552 

.20345 

4.91516 

30 

31 

.14975 

6.67787 

.16764 

5.96510 

.18564 

5.38677 

.20376 

4.90785 

29 

32 

.15005 

6.66463 

.16794 

5.95448 

.18594 

5.37805 

..20406 

4.90056 

28 

33 

.15034 

6.65144 

.16824 

5.94390 

.18624 

5.36936 

.20436 

4.89330 

27 

34 

.15064 

6.63831 

.16854 

5.93335 

. 18654 

5.36070 

.20466 

4.88605 

26 

35 

.15094 

6.62523 

.16884 

5.92283 

.18684 

5.35206 

.20497 

4.87882 

25 

36 

.15124 

6.61219 

.16914 

5.91236 

.18714 

5.34345 

.20527 

4.87162 

24 

37 

.15153 

6.59921 

.16944 

5.90191 

.18745 

5.33487 

.20557 

4.8C444 

23 

38 

.15183 

6.58627 

.16974 

5.89151 

.18775 

5.32631 

.20588 

4.85727 

22 

39 

.15213 

6.57339 

.17004 

5.88114 

.18805 

5.31778 

.20618 

4.85013 

21 

40 

.15243 

6 . 56055 

.17033 

5.87080 

.18835 

5.30928 

.20648 

4.84300 

20 

41 

.15272 

6.54777 

.17063 

5.86051 

.18865 

5.30080 

.20679 

4.83590 

19 

42 

.15302 

6 . 53503 

.17093 

5.85024 

.18895 

5.29235 

.20709 

4.82882 

18 

43 

.15332 

6.52234 

.17123 

5.84001 

.18925 

5.28393 

.20739 

4.82175 

17 

44 

.15362 

6.50970 

.17153 

5.82982 

.18955 

5.27553 

.20770 

4.81471 

16 

45 

.15391 

6.49710 

.17183 

5.81966 

.18986 

5.26715 

.20800 

4.80769 

15 

46 

. 15421 

6.48456 

.17213 

5.80953 

.19016 

5.25880 

.20830 

4.80068 

14 

47 

.15451 

6.47206 

.17243 

5.79944 

.19046 

5.25048 

.20861 

4.79370 

13 

48 

.15481 

6:45961 

.17273 

5.78938 

.19076 

5.24218 

.20891 

4.78673 

12 

49 

.15511 

6.44720 

.17303 

5.77936 

.19106 

5.23391 

I .20921 

4.77978 

11 

50 

.15640 

6.43484 

.17333 

5.76937 

.19136 

5.22566 

.20952 

4.77286 

10 

51 

.15570' 

6.42253 

.17363 

5.75941 

.19166 

5.21744 

.20982 

4.76595 

9 

52 

.15600 

6.41026 

.17393 

5.74949 

.19197 

5.20925 

.21013 

4.75906 

8 

53 

. 15630 

6.39804 

.17423 

5.73960 

.10227 

5.20107 

.21043 

4.75219 

7 

54 

.15660 

6.38587 

.17453 

5.72974 

.19257 

5.19293 

.21073 

4.74534 

6 

55 

.15689 

6.37374 

.17483 

5.71992 

.19287 

5.18480 

.21104 

4.73851 

5 

| 56 

.15719 

6.36165 

.17513 

5.71013 

.19317 

5.17671 

.21134 

4.73170 

4 

57 

.15749 

6.34961 

.17543 

5.70037 

.19347 

5.16863 

.21164 

4.72490 

o 

| 58 

. 15779 

6.33761 

.17573 

5.69064 

.19378 

5.16058 

.21195 

4.71813 

l 

! 59 

.15809 

G.  32566 

.17603 

5.68094 

.19408 

5.15256 

.21225 

4.71137 

1 

60 

. 15838 

6.31375 

.17633 

5.67128 

.19438 

5.14455 

.21256 

4.70463 

0 

Cotang 

Tang 

Cotang 

Tang 

Cotang 

Tang 

Cotang 

Tang 

/ 

J / 

81° 

80°  I 

79°  1 

78° 

TABLE  II.  TANGENTS  AND  COTANGENTS. 


143 


12° 

13° 

14°  1 

15° 

Tang 

Cotang 

Tang 

Cotang 

Tang 

Cotang 

Tang  | Cotang 

f 

0 

,.21256 

4.70463 

.23087 

4.33148 

.24933 

4.01078 

.26795 

3.73205 

60 

1 

.21286 

4.69791 

.23117 

4.32573 

.24964 

4.00582 

.26826 

3.72771 

59 

2 

.21316 

4.69121 

.23148 

4.32001 

.24995 

4.00086 

.26857 

3.72338 

58 

3 

.21347 

4.68452 

.23179 

4.31430 

.25026 

3.99592 

. /SOoOO 

3.71907 

57 

4 

.21377 

4.67786 

.23209 

4.30860 

.25056 

3.99099 

.26920 

3.71476 

56 

5 

.21408 

4.67121 

.23240 

4.30291 

.25087 

3.98607 

.26951 

3.71046 

55 

6 

.21438 

4.66458 

.23271 

4.29724 

.25118 

3.98117 

.26982 

3.70616 

54 

7 

.21469 

4.65797 

.23301 

4.29159 

.25149 

3.97627 

.27013 

3.70188 

53 

8 

.21499 

4.65138 

.23332 

4.28595 

.25180 

3.97139 

.27044 

3.69761 

52 

9 

.21529 

4.64480 

.23363 

4.28032 

.25211 

3.96651 

.27076 

3.69335 

51 

10 

.21560 

4.63825 

.23393 

4.27471 

.25242 

3.96165 

.27107 

3.68909 

50 

11 

.21590 

4.63171 

.23424 

4.26911 

.25273 

3.95680 

.27138 

3.68485 

49 

12 

.21621 

4.62518 

.23455 

4.26352 

.25304 

3.95196 

.27169 

3.68061 

48 

13 

.21651 

4.61868 

.23485 

4.25795 

.25335 

3.94713 

.27201 

3.67638 

47 

14 

.21682 

4.61219 

.23516 

4.25239 

.25366 

3.94232 

.27232 

3.67217 

46 

15 

.21712 

4.60572 

.23547 

4.24685 

.25397 

3.93751 

.27263 

3.66796 

45 

16 

.21743 

4.59927 

.23578 

4.24132 

.25428 

3.93271 

.27294 

3.66376 

44 

17 

.21773 

4.59283 

.23608 

4.23580 

.25459 

3.92793 

.27326 

3.65957 

43 

18 

.21804 

4.58641 

.23639 

4.23030 

.25490 

3.92316 

. 27357 

3.65538 

42 

19 

.21834 

4.58001 

.23670 

4.22481 

.25521 

3.91839 

.27388 

3.65121 

41 

20 

.21864 

4.57363 

.23700 

4.21933 

.25552 

3.91364 

.27419 

3.64705 

40 

21 

.21895 

4.56726 

.23731 

4.21387 

.25583 

3.90890 

.27451 

3.64289 

39 

22 

.21925 

4.56091 

.23762 

4.20842 

.25614 

3.90417 

.27482 

3.63874 

38 

23 

.21956 

4.55458 

.23793 

4.20298 

.25645 

3.89945 

.27513 

3.63461 

37 

24 

.21986 

4.54826 

.23823 

4.19756 

.25676 

3.89474 

.27545 

3.63048 

36 

25 

.22017 

4.54196 

.23854 

4.19215 

.25707 

3.89004 

.27576 

3.62636 

35 

26 

.22047 

4.53568 

.23885 

4.18675 

.25738 

3.88536 

.27607 

3.62224 

34 

27 

.22078 

4.52941 

.23916 

4.18137 

.25769 

3.88068 

.27638 

3.61814 

33 

28 

.22108 

4.52316 

.23946 

4.17600 

.25800 

3.87601 

.27670 

3.61405 

32 

29 

.22139 

4.51693 

.23977 

4.17064 

.25831 

3.87136 

.27701 

3.60996 

31 

30 

.22169 

4.51071 

.24008 

4.16530 

.25862 

3.86671 

.27732 

3 60588 

30 

31 

.22200 

4.50451 

.24039 

4.15997 

.25893 

3.86208 

.27764 

3.60181 

29 

32 

.22231 

4.49832 

.24069 

4.15465 

.25924 

3.85745' 

.27795 

3.59775 

28 

33 

.22261 

4.49215 

.24100 

4.14934 

.25955 

3.85284 

.27826 

3.59370 

27 

34 

.22292 

4.48600 

.24131 

4.14405 

.25986 

3.84824 

.27858 

3.58966 

26 

35 

.22322 

4.47986 

.24162 

4.13877 

.26017 

3.84364 

.27889 

3.58562 

25 

36 

.22353 

4.47374 

.24193 

4.13350 

.26048 

3.83906 

.27921 

3.58160 

24 

37 

.22383 

4.46764 

.24223 

4.12825 

26079 

3.83449 

.27952 

3.57758 

23  ■ 

38 

.22414 

4.46155 

.24254 

4.12301 

.26110 

3.82992 

.27983 

3.57357 

22 

39 

.22444 

4.45548 

: 24285 

4.11778 

.26141 

3.82537 

.28015 

3.56957 

21 

40 

.22475 

4.44942 

.24316 

4.11256 

.26172 

3.82083 

.28046 

3.56557 

20 

41 

.22505 

L 4.44338 

.24347 

4.10736 

.26203 

3.81630 

.28077 

3.56159 

19 

42 

.22536 

4.43735 

.24377 

4.10216 

.26235 

3.81177 

.28109 

3 . 55761 

18 

43 

.22567 

4.43134 

.24408 

4.09699 

.26266 

3.80726 

.28140 

3.55364 

17 

44 

.22597 

4.42534 

.24439 

4.09182 

.26297 

3.80276 

.28172 

3.54968 

16 

45 

.22628 

4.41936 

.24470 

4.08666 

.26328 

3.79827 

.28203 

3.54573 

15 

46 

.22658 

4.41340 

.24501 

4.08152 

.26359 

3.79378 

.28234 

3.54179 

14 

47 

.22689 

4.40745 

.24532 

4.07639 

.28390 

3.78931 

. 28266 

3.53785 

13 

48 

.22719 

4.40152 

.24562 

4.07127 

.26421 

3.78485 

.28297 

3.53393 

12 

49 

.22750 

4.39560 

.24593 

4.06616 

.26452 

3.78040 

.28329 

3.53001 

11 

50 

.22781 

4.38969 

.24624 

4.06107 

.26483 

3.77595 

.28360 

3.52609 

10 

51 

.22811 

4.38381 

.24655 

4.05599 

.26515 

3.77152 

.28391 

3.82219 

9 

52 

.22842 

4.37793 

.24686 

4.05092 

.26546 

3.76709 

.28423 

3.51829 

8 

53 

.22872 

4.37207 

.24717 

4.04586 

.26577 

3.76268 

.28454 

3.51441 

7 

54 

.22903 

4.36623 

.24747 

4.04081 

.26608 

3.75828 

.28486 

3.51053 

6 

55 

.22934 

4.36040 

.24778 

4.03578 

.26639 

3.75388 

.28517 

3.50666 

5 

56 

.22964 

4.35459 

.24809 

4.03076 

.26670 

3.74950 

.28549 

3.50279 

4 

57 

.22995 

4.34879 

.24840 

4.02574 

.26701 

3.74512 

.28580 

3.49894 

3 

58 

.23026 

4.34300 

.24871 

4.02074 

.26733 

3.74075 

.28612 

3.49509 

2 

59 

.23056 

4.33723 

.24902 

4.01576 

.26764 

3.73640 

.28643 

3.49125 

1 

(50 

.23087 

4.33148 

.24933 

4.01078 

_L26795 

3.73205  ' 

.28675 

3.48741 

0 

/ 

Cotang 

Tang 

Cotang 

Tang 

, Cotang 

Tang 

Cotang 

Tang 

7 

77° 

76° 

75° 

74° 

144 


TABLE  II.  TANGENTS  AND  COTANGENTS. 


16° 

17° 

CO 

| 19° 

Tang 

1 Cotang 

Tang 

Cotang 

Tang 

Cotang 

Tang 

Cotang 

/ 

0 

.28675 

3.48741 

.30573 

3.27085 

.32492 

3.07768 

.34433 

2.90421 

60 

1 

.28706 

3.48359 

.30605 

3.26745 

.32524 

3.07464 

.34465 

2.90147 

59 

2 

.28738 

3.47977 

.30637 

3.26406 

! .32556 

3.07160 

.34498 

2.89873 

58 

3 

.28769 

3.47596 

.30669 

3.26067 

; .32588 

3.06857 

.34530 

2.89600 

57 

4 

.28800 

3.47216 

.30700 

3.25729 

.32621 

3.06554 

.34563 

2.89327 

56 

5 

.28832 

3.46837 

.30732 

3.25392 

.32653 

3.06252 

.34596 

2.89055 

55 

C 

.28864 

3.46458 

.30764 

3.25055 

.32685 

3.05950 

.34628 

2.88783 

54 

7 

.28895 

3.46080 

.30796 

3.24719 

.32717 

3.05649 

.34661 

2.88511 

53 

8 

.28927 

3.45703 

.30828 

3.24383 

.32749 

3.05349 

.34693 

2.88240 

52 

9 

.28958 

3.45327 

.30860 

3.24049 

.32782' 

3.05049 

.34726 

2.87970 

51 

10 

.28990 

3.44951 

.30891 

3.23714 

.32814 

3.04749 

.34758 

2 87700 

50 

11 

.29021 

3.44576 

.30923 

3.23381 

.32846 

3.04450 

.34791 

2.87430 

49 

12 

.29053 

3.44202 

.30955 

3.23048 

.32878 

3.04152 

.34824 

2.87161 

48 

13 

.29084 

3.43829 

.30987 

3.22715 

.32911 

3.03854 

.34856 

2.86892 

17 

14 

.29116 

3.43456 

.31019 

3.22384 

.32943 

3.03556 

.34889 

2.86624 

46 

15 

.29147 

3.43084 

.31051 

.3.22053 

.32975 

3.03260 

.34922 

2.86356 

45 

16 

.29179 

3.42713 

.31083 

3.21722 

.33007 

3.02963 

.34954 

2.86089 

44 

17 

.29210 

3.42343 

.31115 

3.21392 

.33040 

3.02667 

.34987 

2.85822 

43 

IS 

.29242 

3.41973 

.31147 

3.21063 

.33072 

3.02372 

.35020 

2.85555 

42 

19 

.29274 

3.41604 

.31178 

3.20734 

.33104 

3.02077 

.35052 

2.85289 

41 

20 

.29305 

3.41236 

.31210 

3.20406 

.33136 

3.01783 

.35085 

2.85023 

40 

21 

.29337 

3.40869 

.31242 

3.20079 

.33169 

3.01489 

.35118 

2.84758 

39 

22 

.29368 

3.40502 

.31274 

3.19752 

.33201 

3.01196 

.35150 

2.84494 

38 

23 

.29400 

3.40136 

.31306 

3.19426 

.33233 

3.00903 

.35183 

2.84229 

37 

24 

.29432 

3.39771 

.31338 

3.19100 

.33266 

3.00611 

.35216 

2.83965 

36 

25 

.29463 

3.39406 

.31370 

3.18775 

.33298 

3.00319 

.35248 

2.83702 

35 

20 

.29495 

3.39042 

.31402 

3.18451 

.33330 

3.00028 

.35281 

2.83439 

34 

27 

.29526 

3.38679 

.31434 

3.18127 

.33363 

2.99738 

.35314 

2.83176 

33 

28 

.29558 

3.38317 

.31466 

3.17804 

.33395 

2.99447 

.35346 

2.82914 

32 

29 

.29590 

3.37955 

.31498 

3.17481 

.33427 

2.99158 

.35379 

2.82653 

31 

30 

.29621 

3.37594 

.31530 

3.17159 

.33460 

2.98868 

.35412 

2.82391 

30 

31 

.29653 

3.37234 

.31562 

3.16838 

.33492 

2.98580 

.35445 

2.82130 

29 

32 

.29685 

3.36875 

.31594 

3.16517 

.33524 

2.98292 

.35477 

2.81870 

28 

33 

.29716 

3.36516 

.31626 

3.16197 

.33557 

2.98004 

.35510 

2.81610 

27 

34 

.29748 

3.36158 

.31658 

3.15877 

.33589 

2.97717 

.35543 

2.81350 

26 

35 

.29780 

3.35800 

.31690 

3.15558 

.33621 

2.97430 

.35576 

2.81091 

25 

36 

.29811 

3.35443 

.31722 

3.15240 

.33654 

2.97144 

.35608 

2.80833 

24 

37 

.29843 

3.35087 

.31754 

3.14922- 

.33686 

2.96858 

.35641 

2.80574 

23' 

38 

.29875 

3.34732 

.31786 

3.14605 

.33718 

2.96573 

.35674 

2.80316 

22! 

39 

.29906 

3.34377 

.31818 

3.14288 

.33751 

2.96288 

.35707 

2.80059 

21 

40 

.29938 

3.34023 

.31850 

3.13972 

.33783 

2.96004 

.35740 

2.79802 

20 

41 

.29970 

3.33670 

.31882 

3.13656 

.33816 

2.95721 

.35772 

2.79545 

19 

^2 

.30001 

3.33317 

.31914 

3.13341 

.33848 

2.95437 

.35805 

2.79289 

18 

43 

.30033 

3.32965 

.31946 

3.13027 

.33881 

2.95155 

.35838 

2.79033 

17 

44 

.30065 

3.32614 

.31978 

3.12713 

.33913 

2.94872 

.35871 

2.78778 

16 

45 

.30097 

3.32264 

.32010 

3.12400 

.33945 

2.94591 

.35904 

2.78523 

15 

46 

.30128 

3.31914 

.32042 

3.12087 

.33978 

2.94309 

.35937 

2.78269 

14 

47 

.30160 

3.31565 

.32074 

3.11775 

.34010 

2.94028  1 

.35969 

2.78014 

13 

48 

.30192 

3.31216 

.32106 

3.11464 

.34043 

2.93748 

.36002 

2.77761 

12 

49 

.30224 

3.30868 

.32139 

3.11153 

.34075 

2.93468 

.36035 

2.77507 

11 

50 

.30255 

3.30521 

.32171 

3.10842 

.34108 

2.93189 

.36068 

2.77254 

10 

51 

.30287 

3.30174 

.32203 

3.10532 

.34140 

2.92910 

.36101 

2.77002 

9 

52 

.30319 

3.29829 

.32235 

3.10223 

.34173 

2.92632 

.36134 

2.76750 

8 

53 

.30351 

3.29483 

.32267 

3.09914 

.34205 

2.92354 

.36167 

2.76498 

7 

54 

.30382 

3.29139 

.32299 

3.09606 

.34238 

2.92076 

.36199 

2.76247 

6 

55 

.30414 

3.28795 

.32331 

3.09298 

.34270 

2.91799 

.36232 

2.75996 

5 

56 

.30446 

3.28452 

.32363 

3.08991 

.34303 

2.91523 

.36265 

2.75746 

4 

57 

.30478 

3.28109 

.32396 

3.08685 

.34335 

2.91246 

.36298 

2.75496 

3 

58 

.30509 

3.27767 

.32428 

3.08379 

.34368 

2.90971 

.36331 

2.75246 

2 

59 

.30541 

3.27426 

.32460 

3.08073 

.34400 

2.90696 

.36364 

2.74997 

1 

00 

.30573 

3.27085 

.32492 

3.07768 

.34433 

2.90421 

.36397 

2.74748 

0 

1 , 

Cotang 

Tang 

Cotang 

Tang 

Cotang 

Tang 

Cotang 

Tang 

/ 

L 

CO 

o 

72° 

71° 

70° 

TABLE  II.  TANGENTS  AND  COTANGENTS. 


145 


o 

© 

CM 

21° 

22° 

1 2 

13° 

| Tang1 

| Cotang 

Tang 

Cotang 

Tang 

Cotang  ! 

Tang 

Cotang 

0 

.36397 

2.74748 

.38386 

2.60509 

.40403 

2.47509 

~42447~ 

2.35585 

60 

1 

.36430 

2.74499 

.38420 

2.60283 

.40436 

2.47302 

.42482 

2.35395 

59 

2 

.36463 

2.74251 

.38453 

2.60057 

.40470 

2.47095 

.42516 

2.35205 

58 

3 

.36496 

2.74004 

.38487 

2.59831 

.40504 

2.46888 

.42551 

2.35015 

57 

4 

.36529 

2.73756 

.38520 

2.59606 

.40538 

2.46682 

.42585 

2.34825 

56 

5 

.36562 

2.73509 

.38553 

2.59381 

.40572 

2.46476 

.42619 

2.34636 

55 

G 

.36595 

2.73263 

.38587 

2.59156 

.40606 

2.46270 

.42654 

2.34447 

54 

7 

.36628 

2.73017 

.38620 

2.58932 

.40640 

2.46065 

.42688 

2.34258 

53 

8 

.36661 

2.72771 

.38654 

2.58708 

.40674 

2.45860 

.42722 

2.34069 

52 

9 

.36694 

2. 72526 

.38687 

2.58484 

.40707 

2.45655 

.42757 

2.33881 

51 

10 

.36727 

2.72281 

.38721 

2.58261 

.40741 

2.45451 

.42791 

2.33693 

50 

11 

.36760 

2.72036 

.38754 

2.58038 

.40775 

2.45246 

.42826 

2.33505 

49 

12 

.36793 

2.71792 

.38787 

2.57815 

.40809 

2.45043 

.42860 

2.33317 

48 

13 

.36826 

2.71548 

.38821 

2.57593 

.40843 

2.44839 

.42894 

2.33130 

47 

14 

.36859 

2.71305 

.38854 

2.57371 

.40877 

2.44636 

.42929 

2.32943 

46- 

15 

.36892 

2.71062 

.38888 

2.57150 

.40911 

2.44433 

.42963 

2.32756 

45 

16 

.36925 

2.70819 

.38921 

2.56928 

.40945 

2.44230 

.42998 

2.32570 

44 

17 

.36958 

2.70577 

.38955 

2.56707 

.40979 

2.44027 

.43032 

2.32383 

43 

18 

.36991 

2.70335 

.38988 

2.56487 

.41013 

2.43825 

.43067 

2.32197 

42 

19 

.37024 

2.70094 

.39022 

2.56266 

.41047 

2.43623 

.43101 

2.32012 

41 

20 

.37057 

2.69853 

.39055 

2,56046 

.41081 

2.43422 

.43136 

2.31826 

40 

21 

.37090 

2.69612 

.39089 

2.55827 

.41115 

2.43220 

.43170 

2.31641 

39 

22 

.37123 

2.69371 

.39122 

2.55608 

.41149 

2.43019 

.43205 

2.31456 

38 

23 

.37157 

2.69131 

.39156 

2.55389 

.41183 

2.42819 

.43239 

2.31271 

37 

24 

.37190 

2.68892 

.39190 

2.55170 

.41217 

2.42618 

.4327 '4 

2.31086 

36 

25 

.37223 

2.68653 

.39223 

2.54952 

.41251 

2.42418 

.43308 

2.30902 

35 

26 

.37256 

2.68414 

.39257 

2.54734 

.41285 

2.42218 

.43343 

2.30718 

34 

27 

.37289 

2.68175 

.39290 

2.54516 

.41319 

2.42019 

.43378 

2.30534 

33 

28 

.37322 

2.67937 

.39324 

2.54299 

.41353 

2.41819 

.43412 

2.30351 

32 

29 

.37355 

2.67700 

.39357 

2.54082 

.41387 

2.41620 

.43447 

2.30167 

31 

30 

.37388 

2.67462 

.39391 

2.53865 

.41421 

2.41421 

.43481 

2.29984 

30 

31 

.37422 

2.67225 

.39425 

2.53648 

.41455 

2.41223 

.43516 

2.29801 

29 

32 

.37455 

2.66989 

.39458 

2.53432 

.41490 

2.41025 

.43550 

2.29619 

28 

33 

.37488 

2.66752 

.39492 

2.53217 

.41524 

2.40827 

.43585 

2.29437 

27 

34 

.37521 

2.66516 

.39526 

2.53001 

.41558 

2.40629 

.43620 

2.29254 

26 

35 

.37554 

2.66281 

.39559 

2.52786 

.41592 

2.40432 

.43654 

2.29073 

25 

36 

.37588 

2.66046 

.39593 

2.52571 

.41626 

2.40235 

.43689 

2.28891 

24 

37 

.37621 

2.65811 

.39626 

2.52357 

.41660 

2.40038 

.43724 

2.28710 

23 

38 

.37654 

2.65576 

.39660 

2.52142 

.41694 

2.39841 

.43758 

2.28528 

22 

39 

.37687 

2.65342 

.39694 

2.51929 

.41728 

2.39645 

.43793 

2.28348 

21 

40 

.37720 

2.65109 

.39727 

2.51715 

.41763 

2.39449 

.43828 

2.28167 

20 

41 

.37754  . 

2.64875 

.39761 

2.51502 

.41797 

2.39253 

.43862 

2.27987 

19 

42 

.37787 

2.64642 

.39795 

2.51289 

.41831 

2.39058 

.43897 

2.27806 

18 

43 

.37820 

2.64410 

.39829 

2.51076 

.41865 

2.38863 

.43932 

2.27G26 

17 

44 

.37853 

2.64177 

.39862 

2.50864 

41899 

2.38668 

.43966 

2.27447 

16 

45 

.37887 

2.63945 

.39896 

2.50652 

.41933 

2.38473 

.44001 

2.27267 

15 

46 

.37920 

2*63714 

.39930 

2.50440 

.41968 

2.38279 

.44036 

2.27088 

14 

47 

.37953 

2.63483 

.39963 

2.50229 

.42002 

2.38084 

.44071 

2.26909 

13 

48 

.37986 

2.63252 

.39997 

2.50018 

.42036 

2.37891 

.44105 

2.26730 

12 

49 

.38020 

2.63021 

.40031 

2.49807 

.42070 

2.37697 

.44140 

2.26552 

11 

50 

.38053 

2.62791 

.40065 

2.49597 

.42105 

2.37504 

.44175 

2.26374 

10 

51 

.38086 

2.62561 

.40098 

2.49386 

.42139 

2.37311 

.44210 

2.26196 

9 

52 

.38120 

2.62332 

.40132 

2.49177 

* .42173 

2.37118 

.44244 

2.26018 

8 

53 

.38153 

2.62103 

.40166 

2.48967 

.42207 

2.36925 

.44279 

2.25840 

7 

54 

.38186 

2.61874 

.40200 

2.48758 

.42242 

2.36733 

.44314 

2.25663 

6 

55 

.38220 

2:61646 

.40234 

2.48549 

.42276 

2.36541 

.44349 

2.25486 

5 

56 

.38253 

2.61418 

.40267 

2 48340 

.42310 

2.36349 

.44384 

2.25309 

4 

57 

.38286 

2.61190 

.40301 

2.48132 

.42345 

2.36158 

.44418 

2.25132 

3 

58 

.38320 

2.60963 

.40335 

2.47924 

.42379 

2.35967 

.44453 

2.24956 

2 

59 

.38353 

2.60736 

.40369 

2.47716 

.42413 

2.35776 

.44488 

2 24780 

1 

60 

.38386 

2 . 60509 

.40403 

2.47509 

.42447  - 

2.35585 

.44523  1 

2.24604 

0 

t 

Cotang 

Tang 

Cotang 

Tang 

Cotang 

Tang 

Cotang 

Tang 

/ 

69° 

68G  1 

67° 

66° 

-r-J 

146 


TABLE  II.  TANGENTS  AND  COTANGENTS, 


24° 

25° 

26°  ! 

27° 

Tang 

Cotang 

Tang 

Cotang 

Tang 

Cotang 

Tang 

Cotang 

0 

.44523 

2.24604 

.46631 

2.14451 

.48773 

2.05030 

.50953 

1.96261 

60 

1 

.44558 

2.24428 

.46666 

2.14288 

.48809 

2.04879 

.50989 

1.96120 

159 

2 

.44593 

2.24252 

.46702 

2.14125 

.48845 

2.04728 

.51026 

1 .$597'9 

i58 

3 

.44627 

2.24077 

.46737 

2.13963 

.48881 

2.04577 

.51063 

1.95838 

57 

4 

.44662 

2.23902 

.46772 

2.13801 

.48917 

2.04426 

.51099 

1 .95698 

■56 

5 

.44697 

2.23727 

.46808 

2.13639 

.48953 

2.04276 

.51136 

1.95557 

55 

6 

.44732 

2.23553 

.46843 

2.13477 

.48989 

2.04125 

.51173 

1.95417 

54 

7 

.44767 

2.23378 

.46879 

2.13316 

.49026 

2.0397'5 

.51209 

1.95277 

53  1 

8 

.44802 

2.23204 

.46914 

2.13154 

.49062 

2.03825 

.51246 

1.95137 

52  i 

9 

.44837 

2.23030 

.46950 

2.12993 

.49098 

2.03S75 

.51283 

1.94997 

51  1 

10 

.44872 

2.22857 

.46985 

2.12832 

.49134 

2.03526 

.51319 

1.94858 

50  j 

11 

.44907 

2.22683 

.47021 

2.12671 

.49170 

2.03376 

.51356 

1.94718 

49  ' 

12 

.44942 

2.22510 

.47056 

2.12511 

.49206 

2.03227 

.51393 

1.94579 

48 

13 

.44977 

2.22337 

.47092 

2.12350 

.49242 

2.03078 

.51430 

1.94440 

47 

14 

.45012 

2.22164 

.47128 

2.12190 

.49278 

2.02929 

.51467 

1.94301 

46 

15 

.45047 

2.21992 

.47163 

2.12030 

.49315 

2.02780 

.51503 

1.94162 

45 

16 

.45082 

2.21819 

.47199 

2.11871 

.49351 

2.02631 

.51540 

1.94023 

44 

17 

.45117 

2.21647 

.47234 

2.11711 

.49387 

2.02483 

.51577 

1.93885 

43 

18 

.45152 

2.2147'5 

.47270 

2.11552 

.49423 

2.02335 

.51614 

1.93746 

42 

19 

.45187 

2.21304 

.47305 

2.11392 

.49459 

2.02187 

.51651 

1.93608 

41 

20 

.45222 

2.21132 

.47341 

2.11233 

.49495 

2.02039 

.51688 

1.93470 

40 

21 

.45257 

2.20961 

.47377 

2.11075 

.49532 

2.01891 

.51724 

1.93332 

39 

22 

.45292 

2.20790 

.47412 

2.10916 

.49568 

2.01743 

.51761 

1.93195 

38 

23 

.45327 

2.20619 

.47448 

2.10758 

.49604 

2.01596 

.51798 

1.93057 

37 

24 

.45362 

2.20449 

.47483 

2.10600 

.49640 

2.01449 

.51835 

1.92920 

36 

25 

.45397 

2.20278 

.47519 

2.10442 

.49677 

2.01302 

.51872 

1.92782 

35 

26 

.45432 

2.20108 

.47555 

2.10284 

.49713 

2.01155 

.51909 

1.92645 

34 

27 

.45467 

2.19938 

.47590 

2.10126 

.49749 

2.01008 

.51946 

1.92508 

33 

28 

.45502 

2.19769 

.47626 

2.09969 

.49786 

2.00862 

.51983 

1.92371 

32 

29 

.45538 

2.19599 

.47662 

2.09811 

.49822 

2.00715 

.52020 

1.92235 

31 

30 

.45573 

2.19430 

.47698 

2.09654 

.49858 

2.00569 

.52057 

1.92098 

30 

31 

.45608 

2.19261 

.47733 

2.09498 

.49894 

2.00423 

.52094 

1.91962 

29 

32 

.45643 

2.19092 

.47769 

2.09341 

.49931 

2.00277 

.52131 

1.91826 

28 

33 

.45678 

2.18923 

.47'805 

2.09184 

.49967 

2.00131 

.52168 

1.91690 

27 

34 

.45713 

2.18755 

.47840 

2.09028 

.50004 

1.99986 

.52205 

1.91554 

28 

35 

.45748 

2.18587 

.47876 

2.08872 

.50040 

1.99841 

.52242 

1.91418 

25 

36 

.45784 

2.18419 

.47912 

2.08716 

.50078 

1.99695 

.52279 

1.91282 

24 

37 

.45819 

2.18251 

.47948 

2.08560 

.50113 

1.99550 

.52316 

1.91147 

23 

38 

.45854 

2.18084 

.47984 

2.08405 

.50149 

1.99406 

.52353 

1.91012 

22 

39 

.45889 

2.17916 

.48019 

2.08250 

.50185 

1.99261 

.52390 

1.90876 

21 

40 

.45924 

2.17749 

.48055 

2.08094 

.50222 

1.99116 

.52427 

1.90741 

20 

41 

.45960 

2.17582 

.48091 

2.07939 

.50258 

1.98972 

.52464 

1.90607 

19 

42 

.45995 

2.17416 

.48127 

2.07785 

.50295 

1.98828 

.52501 

1.90472 

18 

43 

.46030 

2.17249 

.48163 

2.07630 

.50331 

1.98684 

.52538 

1.90337 

17 

44 

.46065 

2.17083 

.48198 

2.07476 

.50368 

1.98540 

.52575 

1.90203 

16 

45 

.46101 

2.16917 

.48234 

2.07321 

.50404 

1.98396 

.52613 

1.90069 

15 

46 

.46136 

2.16751 

.48270 

2.07167 

.50441 

1.98253 

.52650 

1.89935 

14 

47 

.46171 

2.16585 

.48306 

2.07014 

.50477 

1.98110 

.52687 

1.89801 

13 

48 

.46206 

2.16420 

.48342 

2.06860 

.50514 

1.97966 

.52724 

1.89667 

12 

49 

.46242 

2.16255 

.48378 

2.06706 

.50550 

1.97823 

.52761 

J. 89533 

li. 

50 

.46277 

2.16090 

.48414 

2.06553 

.50587 

1.97681 

.52798 

1.89400 

10 

51 

.46312 

2.15925 

■ .48450 

2.06400 

.50623 

1.97538 

.52836 

1.89266 

52 

.46348 

2.15760 

.48486 

2.06247 

.50660 

1.97395 

.52873 

1.89133 

0 

a 

53 

.46383 

2.15596 

.48521- 

2.06094 

.50696 

1.97253 

.52910 

1.89000 

7 

54 

.46418 

2.15432 

.48557 

2.05942 

.50733 

1.97111 

.52947 

1.88867 

6 

55 

.46454 

2.15268 

.48593 

2.05790 

.50769 

1.96969 

.52985 

1.88734 

5 

56 

.46489 

2.15104 

.48629 

2.05637 

.50806 

1.96827 

.53022 

1.88602 

4 

57 

.46525 

2.14940 

.48665 

2.05485 

.50843 

1.96685 

.53059 

1.88469 

3 

58 

.46560 

2.14777 

.48701 

2.05333 

.50879 

1.96544 

.53096 

1.88337 

2 

1 

! w 

. 46595 

2.14614 

.48737 

2.05182 

.50916 

1.96402 

.53134 

1.88205 

■ (.0 

.46631 

2.14451  j 

.48773_ 

2.05030 

.50953 

1.96261 

.53171 

1.88073 

0 

17 

Cotang 

Tang 

Cotang  | 

Tang 

Cotang 

Tang 

Cotang 

Tang 

r 

1 ' ! 

65°  1 

64° 

0 

62° 

147 


TABLE  II.  TANGENTS  AND  COTANGENTS. 


to 

00 

29° 

© 

CO 

31° 

f 

Tang 

Cotang 

Tang 

Cotang 

Tang 

Cotang 

Tang 

Cotang 

0 

.53171 

1.88073 

.55431 

1.80405 

.57735 

1.73205 

.60086 

1.66428 

60 

1 

.53208 

1.87941 

.55469 

1.80281 

.57774 

1.73089 

.60126 

1.66318 

59 

2 

.53246 

1.87809 

.55507 

1.80158 

.57813 

1.72973 

.60165 

1.66209 

58 

3 

.53283 

1.87677 

.55545 

1.80034 

.57851 

1.72857 

.60205 

1 . 66099 

57  ! 

4 

.53320 

1.87546 

.55583 

1.79911 

.57890 

1.72741 

.60245 

1.65990 

56  ! 

5 

.53358 

1.87415 

.55621 

1.79788 

.57929 

1.72625 

.60284 

1.65881 

55  ! 

6 

.53395 

1.87283 

.55659 

1.79665 

.57968 

1.72509 

.60324 

1.65772 

54  | 

7 

.53432 

1.87152 

.55697 

1.79542 

.58007 

1.72393 

.60364 

1.65663 

53  i 

8 

.53470 

1.87021 

.55736 

1.79419 

.58046 

1.72278 

.60403 

1.65554 

9 

.53507 

1.86891 

.55774 

1.79296 

.58085 

1.72163 

.60443 

1.65445 

51 

10 

.53545 

1.86760 

.55812 

1.79174 

.58124 

1.72047 

.60483 

1.65337 

50 

11 

.53582 

1.86630 

.55850 

1.79051 

.58162 

1.71932 

.60522 

1.65228 

49 

12 

.53620 

1.86499 

.55888 

1.78929 

.58201 

1.71817 

.60562 

1.65120 

48 

13 

.53657 

1.86369 

.55926 

1.78807 

.58240 

1.71702 

.60602 

1.65011 

47 

14 

.53694 

1.86239 

.55964 

1.78685 

.58279 

1.71588 

.60642 

1.64903 

48 

15 

.53732 

1.86109 

.56003 

1.78563 

.58318 

1.71473 

.60681 

1.64795 

45 

16 

.53769 

1.85979 

.56041 

1.78441 

.58357 

1.71358 

.60721 

1 . 64687 

44 

17 

.53807 

1.85850 

.56079 

1.78319 

.58396 

1.71244 

.60761 

1.64579 

43 

18 

.53844 

1.85720 

.56117 

1.78198 

.58435 

1.71129 

.60801 

1.64471 

42 

19 

.53882 

1.85591 

.56156 

1.78077 

.58474 

1.71015 

.60841 

1.64363 

41 

20 

.53920 

1.85462 

.56194 

1.77955 

.58513 

1.70901 

.60881 

1.64256 

40 

21 

.53957 

1.85333 

.56232 

1.77834 

.58552 

1.70787 

.60921 

1.64148 

39 

22 

.53995 

1.85204 

.56270 

1.77713 

.58591 

1.70673 

.60960 

1.64041 

38 

23 

.54032 

1.85075 

.56309 

1.77592 

.58631 

1.70560 

.61000 

1.63934 

37 

24 

.54070 

1.84946 

.56347 

1.77471 

.58670 

1.70446 

.61040 

1.68826 

36 

25 

.54107 

1.84818 

.56385 

1.77351 

.58709 

1.70332 

.61080 

1.63719 

35 

26 

.54145 

1.84689 

.56424 

1.77230 

.58748 

1.70219 

.61120 

1.63612 

34 

27 

.54183 

1.84561 

.56462 

1.77110 

.58787 

1.70106 

.61160 

1.63505 

33 

28 

.54220 

1.84433 

.56501 

1,76990 

.58826 

1 . 69992 

.61200 

1.63398 

32 

29 

.54258 

1.84305 

.56539 

1.76869 

.58865 

1.69879 

.61240 

1.63292 

31 

30 

.54296 

1.84177 

.56577 

1.76749 

.58905 

1.69766 

.61280 

1.63185 

30 

31 

.54333 

1.84049 

.56616 

1.76629 

.58944 

1.69653 

.61320 

1.63079 

29 

32 

.54371 

1.83922 

.56654 

1.76510 

.58983 

1.69541 

.61360 

1.62972 

28 

33 

.54409 

1.83794 

.56693 

1.76390 

.59022 

1.69428 

.61400 

1.62866 

27 

34 

.54446 

1.83667 

.56731 

1.76271 

.59061 

1.69316 

.61440 

1.62760 

26 

35 

.54484 

1.83540 

.56769 

1.76151 

.59101 

1.69203 

.61480 

1.62654 

25 

36 

.54522 

1.83413 

.56808 

1.76032 

i .59140 

1.69091 

.61520 

1.62548 

24 

37 

.54560 

1.83286 

.56846 

1.75913 

.59179 

1.68979 

.61561 

1.62442 

23 

38 

.54597 

1.83159 

.56885 

1.75794 

! .59218 

1.68866 

.61601 

1.62336 

22 

39 

.54635 

1.83033 

.56923 

1.75675 

| .59258 

1.68754 

.61641 

1.62230 

21 

40 

.54673 

1.82906 

.56962 

1.75556 

| .59297 

1.68843 

.61681 

1.62125 

20 

41 

.54711 

1.82780 

.57000 

1.75437 

.59336 

1.68531 

.61721 

1.62019 

19 

42 

.54748 

1.82654 

.57039 

1.75319 

.59376 

1.68419 

.61761 

1.61914 

18 

43 

.54786 

1.82528 

.57078 

1.75200 

.59415 

1.68308 

.61801 

1.61808 

17 

44 

.54824 

1.82402 

.57116 

1.75082 

.59454 

1.68196 

.61842 

1.61703 

16 

45 

.54862 

1.82276 

.57155 

1.74964 

.59494 

1.68085 

.61882 

1.61598 

15 

46 

.54900 

1.82150 

.57193 

1.74846 

.59533 

1.67974 

.61922 

1.61493 

14 

47 

.54938 

1.82025 

.57232 

1.74728 

.59573 

1.67863 

.61962 

1.61388 

13 

48 

.54975 

1.81899 

.57271 

1.74610 

.59612 

1.67752 

.62003 

1.61283 

12 

49 

.55013 

1.81774 

.57309 

1.74492 

.59651 

1.67641 

.62043 

1.61179 

jll 

50 

. 55051 

1.81649 

.57348 

1.74375 

.59691 

1.67530 

.62083 

1.61074 

10 

51 

.55089 

1.81524 

.57386 

1.74257 

.59730 

1.67419 

.62124 

1.60970 

9 

52 

.55127 

1.81399 

.57425 

1.74140 

.59770 

1.67309 

.62164 

1.60865 

8 

53 

.55165 

1.81274 

.57464 

1.74022 

.59809 

1:67198 

.62204 

1.60761 

7 

54 

.55203 

1.81150 

.57503 

1.73905 

.59849 

1.67088 

.62245 

1.60657 

6 

55 

.55241 

1.81025 

.57541 

1.73788 

.59888 

1.66978 

.62285 

1.60553 

5 

56 

.55279 

1.80901 

.57580 

1.73671 

.59928 

1.66867 

.62325 

1.60449 

4 

57 

.55317 

1.80777 

.57619 

1.73555 

.59967 

1.66757 

.62366 

1 .60345 

3 

58 

. 55355 

1.80653 

.57657 

1.73438 

.60007 

1.66647 

.62406 

1.60241 

2 

59 

.55393 

1 . 80529 

.57696 

1.73321 

.60046 

1 . 66538 

.62446 

1.60137 

1 

60 

.55431 

1.80405 

.57735 

1.73205 

.60086 

1.66428 

.62487 

1.60033 

0 

Cotang 

Tang 

Cotang 

! Tang 

Cotang 

Tang 

Cotang  | 

Tang 

/ 

61° 

1 60° 

i 59° 

58° 

148 


TABLE  II.  TANGENTS  AND  COTANGENTS. 


’ 

32°  1 

o 

CO 

CO 

o 

CO 

lO 

00 

Tang 

Cotang 

Tang 

Cotang 

Tang 

Cotang  ! 

Tang 

Cotang 

0 

.62487 

1.60033 

.64941 

1.53986 

'.6745T 

1.48256 

.70021 

1.42815 

60 

1 

.62527 

1.59930 

.64982 

1.53888 

.67493 

1.48163 

.70064 

1.42726 

59 

2 

.62568 

1.59826 

. 65024 

1.53791 

.67536 

1.48070 

.70107 

1.42638 

58 

3 

.62608 

1.59723 

.65065 

1.53693 

.67578 

1.47977 

.70151 

1.42550 

57 

4 

.62649 

1.59620 

.65106 

1.53595 

.67620 

1.47885 

.70194 

1.42462 

56 

5 

.62689 

1.59517 

.65148 

1.53497 

.67663 

1.47792 

.70238 

1.42374 

55 

6 

.62730 

1.59414 

.65189 

1.53400 

.67705 

1.47699 

.70281 

1.42286 

54 

. 7 

.62770 

1.59311 

.65231 

1.53302 

.67748 

1.47607 

.70325 

1.42198 

53 

8 

.62811 

1.59208 

.65272 

1.53205 

.67790 

1.47514 

.70388 

1.42110 

52 

9 

.62852 

1.59105 

.65314 

1.53107 

.67832 

1.47422 

.70412 

1.42022 

51 

10 

.62892 

1.59002 

.65355 

1.53010 

.67875 

1.47330 

.70455 

1.41934 

50 

11 

.62933 

1.58900 

.65397 

1.52913 

.67917 

1.47238 

.70499 

1.41847 

49 

12 

.62973 

1 .58797 

.65438 

1.52816 

.67960 

1.47146 

.70542 

1.41759 

48 

13 

.63014 

1.58695 

.65480 

1.52719 

.68002 

1.47053 

.70586 

1.41672 

47 

14 

.63055 

1.58593 

.65521 

1.52622  , 

.68045 

1.46962 

.70629 

1.41584 

46 

15 

.63095 

1.58490 

.65563 

1.52525 

.68088 

1.46870 

.70673 

1.41497 

45 

16 

.63136 

1.58388 

.65604 

1.52429  1 

.68130 

1.46778 

.70717 

1.41409 

44 

ir 

.63177 

1.58286 

.65646 

1.52332 

.68173 

1.46686 

.7'0760 

1.41322 

43 

18 

.63217 

1.58184 

.65688 

1.52235 

.68215 

1.46595 

.70804 

1.41235 

42 

19 

.63258 

1.58083 

.65729 

1.52139 

.68258 

1.46503 

.70848 

1.41148 

41 

20 

.63299 

1.57981 

.65771 

1.52043 

.68301 

1.46411 

.70891 

1.41061 

40 

21 

.63340 

1.57879 

.65813 

1.51946 

.68343 

1.46320 

.70935 

1.40974 

39 

22 

.63380 

1.57778 

.65854 

1.51850 

.68386 

1.46229 

.70979 

1.40887 

38 

23 

.63421 

1.57676 

.65896 

1.51754 

.68429 

1.46137 

.71023 

1.40800 

37 

24 

.63462 

1.57575 

.65938 

1.51658 

.68471 

1.46046 

.71066 

1.40714 

36 

25 

.63503 

1.57474 

.65980 

1.51562 

.68514 

1.45955 

.71110 

1.40627 

35 

26 

.63544 

1.57372 

.66021 

1.51466 

.68557 

1.45864 

.71154 

1.40540 

34 

27 

.63584 

1 . 57271 

.66063 

1.51370 

.68600 

1.45773 

.71198 

1.40454 

33 

28 

.63625 

1.57170 

.66105 

1.51275 

.68642 

1.45682 

.71242 

1.40367 

32 

29 

.63666 

1.57069 

.66147 

1.51179 

.68685 

1.4559,2 

.71285 

1.40281 

31 

30 

.63707 

1.56969 

.66189 

1.51084 

.68728 

1.45501 

.71329 

1.40195 

30 

31 

.63748 

1.56868 

.66230 

1.50988 

.68771 

1.45410 

.71373 

1.40109 

29 

32 

.63789 

1.56767 

.66272 

1.50893 

.68814 

1.45320 

.71417 

1.40022 

28 

33 

.63830 

1.56667 

.66314 

1.50797 

.68857 

1.45229 

.71461 

1.39936 

27 

34 

.63871 

1.56566 

.66356 

1.50702 

.68900 

1.45139 

.71505 

1.39850 

26 

35 

.63912 

1.56466 

.66398 

1.50607 

.68942 

1.45049 

.71549 

1.39764 

25 

36 

.63953 

1.56366 

.66440 

1.50512 

.68985 

1.44958 

.71593 

1.39679 

24 

37 

.63994 

1.56265 

.66482 

1.50417 

.69028 

1.44868 

.71637 

1.39593 

23 

38 

.64035 

1.56165 

.66524 

1.50322 

.69071 

1.44778 

.71681 

1.39507 

22 

39 

.64076 

1.56065 

.66566 

1.50228 

.69114 

1.44688 

.71725 

1.39421 

21 

40 

.64117 

1.55966 

.66608 

1.50133 

.69157 

1.44598 

.71769 

1.39336 

20 

41 

.64158 

1.55866 

.66650 

1.50038 

.69200 

1.44508 

.71813 

1.39250 

19 

42 

.64199 

1.55766 

.66692 

1.49944 

.69243 

1.44418 

.71857 

1.39165 

18 

43 

.64240 

1.55666 

.66734 

1.49849 

.69286 

1.44329 

.71901 

1.39079 

17 

44 

.64281 

1.55567 

.66776 

1.49755 

.69329 

1.44239 

.71946 

1.38994 

16 

45 

.64322 

1.55467 

.66818 

1.49661 

.69372 

1.44149 

.71990 

1.38909 

15 

46 

.64363 

1.55368 

’ .66860 

1.49566 

.69416 

1.44060 

.72034 

1.38824 

14 

47 

.64404 

1.55269 

.66902 

1.49472 

.69459 

1.43970 

.72078 

1.38738 

13  i 

48 

.(T4446 

1.55170 

.66944 

1.49378 

.69502 

1.43881 

.72122 

1.38653 

12  | 

49 

.64487 

1.55071 

.66986 

1.49284 

.69545 

1.43792 

.72167 

1.38568 

11  1 

50 

.64528 

1.54972 

.67028 

1.49190 

.69588 

1.43703 

.72211 

1.38484 

10 

51 

.64569 

1.54873 

.67071 

1.49097 

.69631 

1.43614 

.72255 

1.38399 

9 

52 

.64610 

1.54774 

.67113 

1.49003 

.69675 

1.43525 

.72299 

1.38314 

8 

53 

.64652 

1.54675 

.67155 

.1.48909 

.69718 

1.43436 

.72344 

1.38229 

7 

54 

.64693 

1.54576 

.67197 

1.48816 

.69761 

1.43347 

.72388 

1.38145 

6 

55 

> .64734 

1.54478 

.67239 

1.48722 

.69804 

1.43258 

.72432 

1.38060 

5 

56 

» .64775 

1.54379 

.67282 

1.48629 

.69847 

1.43169 

.72477 

1.37976 

4 

5? 

' .64817 

1.54281 

.67324 

1.48536 

.69891 

1.43080 

.72521 

1 .37891 

8 

56 

1 .64858 

1.54183 

.67366 

1.48442 

.69934 

1.42992 

. 72565 

1 .37807 

2 

re 

1 .64899 

1.54085 

.67409 

1.48349 

.69977 

1.42903 

.72610 

1 .37722 

1 

6C 

) .64941 

1.53986 

.67451 

1.48256 

.70021 

1.42815 

.72654 

1 .37638 

0 

Cotang 

Tang 

Cotang 

J Tang 

Cotang 

Tang 

Cotang 

Tang 

t 

J 57° 

56° 

55° 

II  54° 

TABLE  II.  TANGENTS  AND  COTANGENTS. 


149 


36° 

37° 

o 

QO 

00 

39° 

Tang 

Cotang 

Tang 

Cotang 

Tang 

Cotang 

Tang  | 

Cotang 

0 

.72654 

1.37638 

.75355 

1.32704 

.78129” 

1.27994 

.80978 

1.23490 

60 

1 

.72699 

1.37554 

.75401 

1.32624 

.78175 

1.27917 

.81027 

1.23416 

59 

2 

.72743 

1.37470 

.75447 

1.32514 

.78222 

1.27841 

.81075 

1.23313 

53 

3 

.72788 

1 .37386 

.75492 

1.32464 

.78269 

1.27764 

.81123 

1.23270 

57 

4 

.72832 

1.37302 

.75538 

1.32384 

.78316 

1.27688 

.81171 

1.23196 

66 

5 

.72877 

1.37218 

.75584 

1.32304 

.78363 

1.27611 

.81220 

1.23123 

55 

C 

.72921 

1.37134 

.75629 

1.32224 

.78410 

1.27535 

.81268 

1.23050 

54 

7 

. 72966 

1.37050 

.75675 

1.32144 

.78457 

1.27458 

.81316 

1.22977 

53 

8 

.73010 

1.36967 

.75721 

1.32064 

.78504 

1.27382 

.81364 

1.22904 

52 

9 

.73055 

1.36883 

.75767 

1.31984 

.78551 

1.27306 

.81413 

1.22831 

51 

10 

.73100 

1.36800 

.75812 

1.31904 

.78598 

1.27230 

.81461 

1.22758 

50 

11 

.73144 

1.36716 

.75858 

1.31825 

.78645 

1.27153 

.81510 

1.22685 

49 

12 

.73189 

1.36633 

.75904 

1.31745 

.78692 

1.27077 

.81558 

1.22612 

43 

13 

.73234 

1.36549 

.75950 

1.31666 

.78739 

1.27001 

.81606 

1.22539 

47 

14 

.73278 

1.36466 

.75996 

1.31586 

.78786 

1.26925 

.81655 

1.22467 

46 

15 

.73323 

1.36383 

.76042 

1.31507 

.78834 

1.26849 

.81703 

1.22394* 

45 

16 

.73368 

1.36300 

.76088 

1.31427 

.78881 

1.26774 

.81752 

1.22321 

44 

17 

.73413 

1.36217 

.76134 

1.31348 

.78928 

1.26698 

.81800 

1.22249 

43 

18 

.73457 

1.36134 

.76180 

1.31269 

.78975 

1.26622 

.81849 

1.22176 

42 

19 

.73502 

1.36051 

.76226 

1.31190 

.79022 

1.26546 

.81898 

1.22104 

41 

20 

.73547 

1.35968 

,76272 

1.31110 

.79070 

1.26471 

.81946 

1.22031 

40 

21 

.73592 

1.35885 

.76318 

1.31031 

.79117 

1.26395 

.81995 

1.21959 

39 

22 

.73637 

1.35802 

.76364 

1.30952 

.79164 

1.26319 

.82044 

1.21886 

38 

23 

.73681 

1.35719 

.76410 

1.30873 

.79212 

1.26244 

.82092 

1.21814 

37 

24 

.73726 

1.35637 

.76456 

1.30795 

.79259 

1.26169 

.82141 

1.21742 

30 

25 

.73771 

1.35554 

.76502 

1.30716 

.79306 

1.26093 

.82190 

1.21670 

1.21598 

35 

26 

.73816 

1.35472 

.76548 

1.30637 

.79354 

1.26018 

.82238 

34 

27 

.73861 

1.35389 

.76594 

1.30558 

.79401 

1.25943 

.82287 

1.21526 

33 

28 

.73906 

1.35307 

.76640 

1.30480 

.79449 

1.25867 

.82336 

1.21454 

32 

29 

.73951 

1.35224 

. i OOoO 

1.30401 

.79496 

1.25792 

.82385 

1.21382 

31 

30 

.73996 

1.35142 

.76733 

1.30323 

.79544 

1.25717 

.82434 

1.21310 

30 

31 

.74041 

1.35060 

.76779 

1.30244 

.79591 

1.25642 

.82483 

1.21238 

29 

32 

.74086 

1.34978 

.76825 

1.30166 

.79639 

1.25567 

.82531 

1.21166 

28 

33 

.74131 

1.34896 

.76871 

1.30087 

.79686 

1.25492 

.82580 

1.21094 

27 

34 

.74176 

1.34814 

.76918 

1.30009 

.79734 

1.25417 

.82629 

1.21023 

26 

85 

.74221 

1.34732 

.76964 

1.29931 

.79781 

1.25343 

. 8267’8 

1.20951 

25 

36 

.74267 

1.34650 

.77010 

1.29853 

.79829 

1.25268 

.82727 

1.20879 

24 

37 

.74312 

1.34568 

.77057 

1.29775 

.79877 

1.25193 

.82776 

1.20808 

23 

.38 

.74357 

1.34487 

.77103 

1.29696 

.79924 

1.25118 

.82825 

1.20736 

22 

39 

.74402 

1.34405 

.77149 

1.29618 

.79972 

1.25044 

.82874 

1.20665 

21 

40 

.74447 

1.34323 

.77196 

1.29541 

.80020 

1.24969 

.82923 

1.20593 

20 

41 

.74492 

1.34242 

.77242 

1.29463 

.80067 

1.24895 

.82972 

1.20522 

19 

42 

.74538 

1.34160 

.77289 

1.29385 

.80115 

1.24820 

.83022 

1.20451 

18 

43 

.74583 

1.34079 

.77335 

1.29307 

.80163 

1.24746 

.83071 

1.20379 

17 

44 

.74628 

1.33998 

.77382 

1.29229 

.80211 

1.24672 

.83120 

1.20308 

16 

45 

.74674 

1.33916 

.77428 

1.29152 

.80258 

1.24597 

.83169 

1.20237 

15 

46 

.74719 

1.33835 

.77475 

1 .29074 

.80306 

1.24523 

.83218 

1.20166 

14 

47 

.74764 

1.33754 

.77521 

1.28997 

.80354 

1.24449 

.83268 

1.20095 

13 

48 

.74810 

1.33673 

.77568 

1.28919 

.80402 

1.24375 

.83317 

1.20024 

12 

49 

.74855 

1.33592 

.77615 

1.28842 

.80450 

1.24301 

.83366 

1.19953 

11 

50 

.74900 

1.33511 

.77661 

1.28764 

.80498 

1.24227 

.83415 

1.19882 

10 

51 

.74946 

1.33430 

.77708 

1.28687 

.80546 

1.24153 

.83465 

1.19811 

9 

52 

.74991 

1.33349 

.77754 

1.28610 

.80594 

1.24079 

.83514 

1.19740 

'8 

53 

.75037 

1.33268 

.77801 

1.28533 

.80642 

1.24005 

.83564 

1.19669 

7 

54 

.75082 

1.33187 

.77848 

1.28456 

.80690 

1.23931 

.83613 

1.19599 

6 

55 

.75128 

1.33107 

.77895 

1.28379 

.80738 

1.23858 

.83662 

1.19528 

5 

56 

.75173 

1.33026 

.77941 

1.28302 

.80786 

1.23784 

.83712 

1 .19457 

4 

57 

.75219 

1.32946 

.77988 

1.28225 

.80834 

1.23710 

.83761 

1.19387 

3 

58 

.75264 

1.32865 

.78035 

1.28148 

.80882 

1.23637 

.83811 

1.19316 

2 

59 

.75310 

1.32785 

.78082 

1.28071 

.80930 

1.23563 

.83860 

1.19246 

1 

60 

.75355 

1.32704 

.78129 

1.27994 

.80978 

1.23490 

.83910 

1.19175 

0 

/ 

Cotang 

| Tang 

Cotang 

Tang 

Cotang 

Tang 

Cotang 

Tang 

53° 

52° 

51° 

Cn 

© 

o 

150  TABLE  II.  TANGENTS  AND  COTANGENTS. 


4* 

O 

o 

41° 

42° 

o 

CO 

Tang 

Cotang 

Tang 

Cotang 

Tang  | Cotang 

Tang 

Cotang 

/ 

0 

.83910 

1.19175 

.86929 

1.15037 

.90040 

1.11061 

.93252 

1.07237 

60 

1 

.83960 

1.19105 

.86980 

1.14969 

.90093 

1.10996 

.93306 

1 .07174 

59 

2 

.84009 

1.19035 

.87031 

1.14902 

.90146 

1.10931 

.93360 

1.07112 

58 

3 

.84059 

1 . 18964 

.87082 

1.14834 

.90199 

1.10867 

.93415 

1.07049 

57 

4 

.84108 

1.18894 

.87133 

1.14767 

.90251 

1.10802 

.93469 

1.06987 

56 

5 

.84158 

1.18824 

.87184 

1.14699 

.90304 

1.10737 

.93524 

1.06925 

55 

6 

.84208 

1.18754 

.87236 

1.14632 

.90357 

1. 10672 

.93578 

1.06862 

54 

7 

.84258 

1.18684 

.87287 

1.14565 

.90410 

1.10607 

.93633 

1.06800 

53 

8 

.84307 

1.18614 

.87338 

1.14498 

.90463 

1.10543 

.93688 

1.06738 

52 

9 

.84357 

1.18544 

.87389 

1.14430 

.90516 

1.10478 

.93742 

1.06676 

51 

10 

.84407 

1.18474 

.87441 

1.14363 

.90569 

1.10414 

.93797 

1.06613 

50 

11 

.84457 

1.18404 

.87492 

1.14296 

.90621 

1.10349 

.93852 

1.06551 

40 

i 12 

.84507 

1.18334 

.87543 

1.14229 

.90674 

1.10285 

.93906 

1.06489 

48 

; 13 

.84556 

1.18264 

.87595 

1.14162 

.90727 

1.10220 

.93961 

1 .06427 

47 

14 

.84606 

1.18194 

.87646 

1.14095 

.90781 

1.10156 

.94016 

1.06365 

46 

15 

.84656 

1.18125 

.87698 

1.14028 

.90834 

1.10091 

.94071 

1.06303 

45 

! 16 

.84706 

1.18055 

.87749 

1.13961 

.90887 

1.10027 

.94125 

1.06241 

44 

1 17 

.84756 

1.17986 

.87801 

1.13894 

.90940 

1.09963 

.94180 

1.06179 

43 

! 18 

.84806 

1.17916 

.87852 

1.13828 

.90993 

1.09899 

.94235 

1.06117 

42 

19 

.84856 

1.17846 

.87904 

1.13761 

.91046 

1.09834 

.94290 

1.06056 

41 

20 

. 84906 

1.17777 

.87955 

1.13694 

.91099 

1.09770 

.94345 

1.05994 

40 

21 

.84956 

1.17708 

.88007 

1.13627 

.91153 

1.09706 

.94400 

1.05932 

39 

22 

.85006 

1.17638 

.88059 

1.13561 

.91206 

1.09642 

.94455 

1.05870 

38 

23 

.85057 

1.17569 

.88110 

1.13494 

.91259 

1.09578 

.94510 

1.05809 

37 

24 

.85107 

1.17500 

.88162 

1.13428 

.91313 

1.09514 

.94565 

1.05747 

36 

25 

.85157 

1.17430 

.88214 

1.13361 

.91366 

1.09450 

.94620* 

1.05685 

35 

26 

.85207 

1.17361 

.88265 

1.13295 

.91419 

1.09386 

.94676 

1.05624 

34 

27 

.85257 

1.17292 

.88317 

1.13228 

.91473 

1.09322 

.94731 

1.05562 

33 

28 

.85308 

1.17223 

.88369 

1.13162 

.91526 

1.09258 

.94786 

1.05501 

32 

29 

.85358 

1.17154 

.88421 

1.13096 

.91580 

1.09195 

.94841 

1.05439 

31 

30 

.85408 

1.17085 

.88473 

1.13029 

.91633 

1.09131 

.94896 

1.05378 

30 

31 

.85458 

1.17016 

.88524 

1.12963 

.91687 

1 .09067 

.94952 

1.05317 

29 

32 

.85509 

1.16947 

.88576 

1.12897 

.91740 

1.09003 

.95007 

1.05255 

28 

33 

.85559 

1.16878 

.88628 

1.12831 

.91794 

1.08940 

.95062 

1.05194 

27 

34 

.85609 

1.16809 

.88680 

1.12765 

.93847 

1.08876 

.95118 

1,05133 

26 

35 

.85660 

1.16741 

.88732 

1.12699 

.91901 

1.08813 

.05173 

1.05072 

25 

36 

.85710 

1.16672 

.88784 

1.12633 

.91955 

1.08749 

.95229 

1.05010 

24 

37 

.85761 

1.16603 

.88836 

1.12567 

.92008 

1.08686 

.95284 

1.04949 

23 

38 

.85811 

1 . 16535 

.88888 

1.12501 

.92062 

1.08622 

.95340 

1.04888 

22 

39 

.85862 

1.16466 

.88940 

1.12435 

.92116 

1.08559 

.95395 

1.04827 

21 

40 

.85912 

1.16398 

.88992 

1.12369 

.92170 

1.08496 

.95451 

1.04766 

20 

41 

.85963 

1.16329 

.89045 

1.12303 

.92224 

1.08432 

.95506 

1.04705 

10 

42 

.86014 

1.16261 

.89097 

1.12238 

.92277 

1 .08369 

.95562 

1.04644 

18 

43 

.86064 

1.16192 

.89149 

1.12172 

.92331 

1.08306 

.95618 

1.04583 

17 

44 

.86115 

1.16124 

.89201 

1.12106 

.92385 

1.08243 

.95673 

1.04522 

16 

45 

.86166 

1.16056 

.89253 

1.12041 

.92439 

1.08179 

.95729 

1.04461 

15 

46 

.86216 

1.15987 

.89306 

1.11975 

.92493 

1.08116 

.95785 

1.04401 

14 

47 

. 86267 

1.15919 

.89358 

1.11909 

.92547 

1.08053 

.95841 

1.04340 

13 

48 

.86318 

1.15851 

.89410 

1.11844 

.92601 

1 .07990 

.95897 

1.04279 

12 

49 

.86368 

1.15783 

.89463 

1 11778 

.92655 

1.07927 

.95952 

1.04218 

11 

! 50 

.86419 

1.15715 

.89515 

1.11713 

.92709 

1.07864 

.96008 

1.04158 

10 

1 51 

.86470 

1.15647 

.89567 

1.11648 

.92763 

1.07801 

.96064 

1.04097 

D 

! 52 

.86521 

1 . 15579 

.89620 

1.11582 

.92817 

1.07738 

.96120 

1.04036 

8 

1 53 

.86572 

1.15511 

.89672 

1.11517 

.92872 

1 .07676 

.96176 

1.03976 

7 

54 

.86623 

1 . 15443 

.89725 

1.11452 

.92926 

1.07613 

.96232 

1.03915 

6 

55 

.86674 

1.15375 

.89777 

1.11387 

.92980 

1.07550 

.96288 

1.03855 

5 

56 

.86725 

1.15308 

.89830 

1.11321 

.93034 

1.07487 

.96344 

1 . 03794 

4 

57 

.86776 

1.15240 

.89883 

1.11256 

.93088 

1.07425 

.96400 

1.03734 

3 

58 

.86827 

1.15172 

.89935 

1.11191 

.93143 

1.07362 

.96457 

1 . 03674 

2 

59 

.86878 

1.15104 

.89988 

1.11126 

.93197 

1.07299 

.96513 

1.03613 

i 

60 

.86929 

1 . 15037 

.90040 

1.11061 

.93252 

1.07237 

.96509 

1.03553 

0 

/ 

Cotang 

Tang 

Cotang 

Tang 

Cotang 

Tang 

Cotang 

Tang 

49° 

\ 48* 

47° 

46° 

TABLE  IT.  TANGENTS  AND  COTANGENTS. 


lot 


44° 

, 

44° 

1 

' 

44° 

/ 

Tang 

Cotang 

Tang 

Cotang 

Tang 

Cotang 

0 

.96569 

1.03553 

60 

20 

.97700 

1 .02355 

40 

40 

.98843 

1.01170 

20 

1 

.96625 

1.03493 

59 

21 

. 97756 

1.02295 

39 

41 

.98901 

1.01112 

19 

2 

.96681 

1 .03433 

58 

22 

.97813 

1 .02236 

38 

42 

.98958 

1.01053 

18 

3 

.96738 

1.03372 

57 

23 

.97870 

1.02176 

37 

43 

.99016 

1.00994 

17 

4 

.96794 

1.03312 

56 

24 

.97927 

1.02117 

36 

44 

. 99073 

1.00935 

16 

5 

.96850 

1.03252 

55 

25 

.97984 

1.02057 

35 

45 

.99131 

1.00876 

15 

C 

.96907 

1.03192 

54 

26 

.98041 

1.01998 

34 

46 

.99189 

1.00818 

14 

7 

.96963 

1.03132 

53 

27 

.98098 

1.01939 

33 

47 

.99247 

1.00759 

13 

8 

.97020 

1 .03072 

52 

28 

.98155 

1.01879 

32 

48 

.99304 

1.00701 

12 

9 

.97076 

1.03012 

51 

29 

.98213 

1.01820 

31 

49 

.99362 

1.00642 

11 

10 

.97133 

1.02952 

50 

30 

.98270 

1.01761 

30 

50 

.99420 

1.00583 

10 

11 

.97189 

1.02892 

49 

31 

.98327 

1.01702 

29 

51 

.99478 

1.00525 

9 

12 

.97246 

1.02832 

48 

32 

.98384 

1.01642 

28 

52 

.99536 

1.00467 

8 

13 

.97302 

1.02772 

47 

33 

.98441 

1.01583 

27 

53 

.99594 

1.00408 

7 

14 

.97359 

1.02713 

46 

34 

.98499 

1.01524 

26 

54 

.99652 

1.00350 

6 

15 

.97416 

1.02653 

45 

35 

.98556 

1.01465 

25 

55 

.99710 

1.00291 

5 

16 

.97472 

1.02593 

44 

36 

.98613 

1.01406 

24 

56 

.99768 

1.00233 

4 

17 

.97529 

1.02533 

43 

37 

.98671 

1.01347 

23 

57 

.99826 

1.00175 

3 

18 

.97586 

1.02474 

42 

38 

.98728 

1.01288 

22 

58 

.99884 

1.00116 

2 

19 

.97643 

1.02414 

41 

39 

.98786 

1.01229 

21 

59 

.99942 

1.00058 

1 

20 

.97700 

1.02355 

40 

40 

.98843 

1.01170 

20 

60 

1.00000 

1.00000 

0 

/ 

Cotang 

Tang 

/ 

/ 

Cotang 

Tang 

/ 

/ 

Cotang 

Tang 

/ 

45° 

45° 

45° 

LENGTHS  OF  CIRCULAR  ARCS. 
Radius  = 1. 


Degrees. 

Minutes. 

Seconds. 

1 

0.017  453  293 

0.000  290  888 

0.000  004  848 

2 

.034  906  585 

.000  581  776 

.000  009  696 

3 

.052  359  878 

.000  872  664 

.000  014  544 

4 

.069  813  170 

.001  163  553 

.000  019  393 

5 

.087  266  463 

.001  454  440 

.000  024  241 

6 

.104  719  755 

.001  745  329 

.000  029  089 

7 

.122  173  048 

.002  036  217 

.000  033  937 

8 

.139  626  340 

.002  327  106 

.000  038  785 

9 

.157  079  633 

.002  617  994 

,000  043  633 

10 

.174  532  925 

.002  908  882 

.000  048  481 

152 


TA13LE  111.  MAGNETIC  NEEDLE. 


Table  III. 

DAILY  VARIATION  OF  THE  MAGNETIC  NEEDLE  AT 
PHILADELPHIA,  PA. 


C 

cS 

© 

I 

a 

<3 

i May. 

June. 

Aug. 

Sept. 

Oct. 

> 

o 

£ 

6 

© 

ft 

6a  m. 

+0.6 

/ 

+1.2 

+ .8 

f-2.6 

+3.7 

+3.9 

+4.2 

t 

r4.7 

+3.5 

+1.3 

+1.2 

+0..7 

7 

+ 1.2 

+ 1-9 

- 

-2.9 

-3.5 

+4.7 

+5.0 

+5.4 

^5.7 

+4.5 

+ 1.7 

+1 .7 

r1.0 

8 

+ 2 1 

+2.5 

-3.7 

-4.0 

+4.7 

+5.1 

+5  4 

-5.5 

+4.5 

1+2.2 

+1.9 

-1.4 

9 

+2.5 

+2  5 

K3.4 

-3.4 

+3.2 

+3.8 

+4.0 

-3.7 

+2.S 

+1.9 

+1.6 

-1.6 

10 

+1.6 

+ 1.5 

hi. 8 

-1.5 

+0.8 

+1.2 

+1.5 

-0.6 

-0.1 

+0.8 

+0  4 

-1.1 

11 

-0.8 

-0.2 

-0.6 

-1.1 

-1.9 

-1A 

-1.5 

1 _ 

-2.9 

-3.2 

-0.8 

-1.1 

-0.3 

Noon 

-2.3 

-2  0 

-2.7 

-3.6 

-4.1 

-4.0 

-3.9 

-5.4 

-5.2 

-2.6 

-2.3 

-1.9 

1 

-3.4 

-3.0 

-3.9 

-5.1 

-5.1 

-5.0 

-5.3 

-6.3 

—5.5 

-3.2 

-2.8 

-3.0 

2 

-3.3 

-3.0 

-3.9 

-5.2 

-4.9 

-4.8 

-5.4 

-5.5 

-4.5 

-3.0 

-2.6 

-3.0 

8 

-2.5 

-2.4 

-3.2 

-4.3 

-3.9 

-3.8 

—4.5 

-3.8 

-3.0 

-2.2 

-1.9 

-2.3 

4 

— 1.5 

-1.7 

-2.3 

-3.0 

-2.5 

-2  6 

-3.3 

-2.0 

-1.7 

-1.1 

-1.2 

-1.3 

5 

-0.9  -1.2 

-1.6 

-1.8 

-1.2 

-1.6 

-2.0 

-0.9 

-0.8 

-0.3 

-0.6 

-0.6 

6 

— 0.6^  —0.8 

-1.0 

-0.9 

-0.4 

-0.9 

-1.2 

f 

-0.5 

-0  3 

+0.4 

l-o.l 

!' 

-0.1 

Tlie  above  table,  which  is  taken  from  the  U.  S.  Coast  and 
Geodetic  Survey  Report  for  1881,  gives  the  mean  results  of  five 
years’  observations  of  the  daily  variation  pf  the  magnetic  needle 
at  Philadelphia.  A plus  sign  indicates  a deviation  of  the  north 
end  of  the  needle  to  the  eastward  of  the  magnetic  meridian,  a 
minus  sign  indicates  a deviation  to  the  westward. 

For  other  places  in  the  United  States  the  daily  variation  may 
be  approximately  ascertained  by  multiplying  the  values  for 
Philadelphia  by  the  numbers  taken  from  the  following  supple- 
mentary table.  For  example,  at  a place  in  latitude  45  degrees 


Lat. 

Long. 

70°. 

Long. 

80°. 

Long. 

90°. 

Long. 

100°. 

Long. 

110°. 

Long. 

120°. 

25° 

0.64 

0.64 

0.63 

0 60 

30 

0.71 

0.70 

0.68 

0.66 

0.65 

35 

0.93 

0.86 

0.80 

0 A7 

0.76 

0.74 

40 

1.05 

1.00 

0.93 

0.90 

0.82 

0.80 

45 

1.31 

1.35 

1.20 

1 .05 

0 95 

0.93 

50 

1.50 

1.67 

1.24 

1.14 

and  longitude  95  degrees  the  multiplier  is  1. 13.  In  southern 
latitudes,  moreover,  the  maximum  deviations  occur  about  an 
hour  later  than  in  northern,  and  in  any  particular  case  the 
table  cannot  be  depended  upon  within  one  hour  on  account  of 
minor  irregularities  and  disturbances. 


TABLE  IV.  DEGREES  AND  TIME, 


153 


TO  REDUCE  DEGREES  TO  TIME, 


o 

H. 

M. 

o 

H.  M. 

Degrees. 

Hours. 

Minutes. 

Hours. 

Degrees. 

M. 

o / 

M. 

o / 

/ 

M. 

S. 

' 

M.  S. 

S. 

/ // 

S. 

/ // 

" 

S. 

T. 

// 

S. 

T. 

T. 

//  /// 

T. 

//  /// 

1 

0 

4 

51 

3 

24 

101 

6 44 

1 

15 

1 

0 15 

51 

12  45 

o 

0 

8 

52 

3 

28 

102 

6 48 

i* 

22* 

2 

0 30 

52 

13  0 

3 

0 

12 

53 

3 

32 

103 

6 52 

2 

30 

3 

0 45 

53 

13  15 

4 

0 

16 

54 

3 

36 

104 

6 56 

2* 

37* 

4 

1 0 

54 

13  30 

5 

0 

20 

55 

3 

40 

105 

7 0 

3 

45 

5 

1 15 

55 

13  45 

6 

0 

24 

56 

3 

44 

106 

7 4 

3J 

52* 

6 

1 30 

56 

14  0 

7 

0 

28 

57 

3 

48 

107 

7 8 

4 

60 

7 

1 45 

57 

14  15 

8 

0 

32 

58 

3 

52 

108 

7 12 

4* 

67* 

8 

2 0 

58 

14  30 

9 

0 

36 

59 

3 

56 

109 

7 16 

5 

75 

9 

2 15 

59 

14  45 

10 

0 

40 

60 

4 

0 

110 

7 20 

5* 

82* 

10 

2 30 

60 

15  0 

11 

0 

44 

61 

4 

4 

115 

7 40 

6 

90 

11 

2 45 

61 

15  15 

12 

0 

48 

62 

4 

8 

120 

8 0 

6* 

97* 

12 

3 0 

62 

15  30 

13 

0 

52 

63 

4 

12 

125 

8 20 

7 

105 

13 

3 15 

63 

15  45 

14 

0 

56 

64 

4 

16 

130 

8 40 

7* 

112* 

14 

3 30 

64 

16  0 

15 

1 

0 

65 

4 

20 

135 

9 0 

8 

120 

15 

3 45 

65 

16  15 

16 

1 

4 

66 

4 

24 

140 

9 20 

8* 

127* 

16 

4 0 

66 

16  30 

17 

1 

8 

67 

4 

28 

145 

9 40 

9 

135 

17 

4 15 

67 

16  45 

18 

1 

12 

68 

4 

32 

150 

10  0 

9* 

142* 

18 

4 30 

68 

17  0 

19 

1 

16 

69 

4 

36 

155 

10  20 

10 

150 

19 

4 45 

69 

17  15 

20 

1 

20 

70 

4 

40 

160 

10  40 

10* 

157* 

20 

5 0 

70 

17  30 

21 

1 

24 

71 

4 

44 

165 

11  0 

11 

105 

21 

5 15 

71 

17  45 

22 

1 

28 

72 

4 

48 

170 

11  20 

Hi 

172* 

22 

5 30 

72 

18  0 

23 

1 

32 

73 

4 

52 

175 

11  40 

12 

180 

23 

5 45 

73 

18  15 

24 

1 

36 

74 

4 

56 

180 

12  0 

12* 

187* 

24 

6 0 

74 

18  30 

25 

1 

40 

75 

5 

0 

185 

12  20 

13 

195 

25 

6 15 

75 

18  45 

26 

1 

44 

76 

5 

4 

190 

12  40 

13* 

202* 

26 

6 30 

76 

19  0 

27 

1 

48 

77 

5 

8 

195 

13  0 

14 

210 

27 

6 45 

77 

19  15 

28 

1 

52 

78 

5 

12 

200 

13  20 

14* 

217* 

28 

7 0 

78 

19  30 

29 

1 

56 

79 

5 

16 

205 

13  40 

15 

225 

29 

7 15 

79 

19  45 

30 

2 

0 

80 

5 

20 

210 

14  0 

15* 

232* 

30 

7 30 

80 

20  0 

31 

2 

4 

81 

5 

24 

215 

14  20 

16 

240 

31 

7 45 

81 

20  15 

32 

2 

8 

82 

5 

28 

220 

14  40 

16* 

247* 

32 

8 0 

82 

20  30 

33 

2 

12 

83 

5 

32 

225 

15  0 

17 

255 

33 

8 15 

83 

20  45 

34 

2 

16 

84 

5 

36 

230 

15  20 

17* 

262* 

34 

8 30 

84 

21  0 

35 

2 

20 

85 

5 

40 

235 

15  40 

18 

270 

35 

8 45 

85 

21  15 

36 

2 

24 

’ 86 

5 

44 

240 

16  0 

18* 

277* 

36 

9 0 

86 

21  30 

37 

2 

28 

87 

5 

48 

245 

16  20 

19 

285 

37 

9 15 

87 

21  45 

38 

2 

32 

88 

5 

52 

250 

16  40 

19* 

292* 

38 

9 30 

88 

22  0 

39 

2 

36 

89 

5 

56 

255 

17  0 

20 

300 

39 

9 45 

89 

22  15 

40 

2 

40 

90 

6 

0 

260 

17  20 

20* 

307* 

40 

10  0 

90 

22  30 

41 

2 

44 

91 

6 

4 

270 

18  0 

21 

315 

41 

10  15 

91 

22  45 

42 

2 

48 

92 

6 

8 

280 

18  40 

21* 

3 22* 

42 

10  30 

92 

23  0 

43 

2 

52 

93 

6 

12 

290 

19  20 

22 

330 

43 

10  45 

93 

23  15 

44 

2 

56 

94 

6 

16 

300 

20  0 

22* 

337* 

44 

11  0 

94 

23  30 

45 

• 3 

0 

95 

6 

20 

310 

20  40 

23 

345 

45 

11  15 

95 

23  45 

46 

3 

4 

96 

6 

24 

320 

21  20 

23* 

352* 

46 

11  30 

96 

24  0 

47 

3 

8 

97 

6 

28 

330 

22  0 

24 

360 

47 

11  45 

97 

24  15 

48 

3 

12 

98 

6 

32 

340 

22  40 

48 

12  0 

98 

24  30 

49 

3 

16 

99 

6 

36 

350 

23  20 

49 

12  15 

99 

24  45 

50 

3 

20 

100 

6 

40 

360  | 

24  0 

50 

12  30 

100 

25  0 

TO  REDUCE  TIME  TO  DEGREES. 


154 


TABLE  V.  POLARIS, 


Table  V. 

TIMES  OF  ELONGATION  AND  CULMINATION  OF  POLARIS 

IN  1899. 


Date  in  1899. 

East 

Elongation. 

Upper  Cul- 
mination. 

West 

Elongation- 

Lower  Cul- 
mination. 

h. 

m. 

h. 

m. 

h. 

m. 

li. 

m. 

January 

1 

0 

41.9 

6 

36.7 

12 

31.5 

18 

34.7 

15 

23 

42.7 

5 

41.7 

11 

36.2 

17 

39.4 

February 

1 

22 

35.5 

4 

34.3 

10 

29.1 

16 

32.3 

15 

21 

40.3 

3 

39.0 

9 

33.9 

15 

37.0 

March 

1 

20 

45.1 

2 

43.8 

8 

38.6 

14 

41.8 

15 

19 

50.0 

1 

48.8 

7 

43.5 

13 

46.8 

April 

1 

18 

43.0 

0 

41.7 

6 

36.5 

12 

39.8 

15 

17 

48.0 

23 

42.8 

i 5 

41.5 

11 

44.8 

May 

1 

16 

45.2 

22 

39.9 

i 4 

38.7 

10 

41.9 

15 

15 

50.3 

21 

45.0 

1 3 

43.8 

9 

47.0 

June 

1 

14 

43.6 

20 

38.4 

1 o 

37.1 

8 

40.4 

15 

13 

48.7 

19 

43.5 

1 

42.2 

7 

45.5 

July 

1 

12 

46.1 

18 

40.9 

1 0 

39.6 

6 

42.9 

15 

11 

51.2 

17 

46.0 

! 23 

40.8 

5 

48.0 

August 

1 

10 

44.7 

16 

39.5 

22 

34.3 

4 

41.5 

15 

9 

49.8 

15 

44.6 

21 

39.4 

3 

46.6 

September 

1 

8 

43  2 

14 

38.0 

20 

32.8 

2 

40.0 

15 

7 

48.3 

13 

43.1 

19 

37.9 

1 

45.1 

October 

1 

6 

45.5 

12 

40.3 

18 

35.1 

0 

42.3 

15 

5 

50.5 

11 

45.3 

17 

40.1 

23 

43.4 

November 

1 

4 

43.7 

10 

38.5 

16 

33.3 

22 

36.5 

15 

3 

48.5 

9 

43.3 

15 

38.1 

21 

41.3 

December 

1 

2 

45.5 

8 

40.3 

14 

35.1 

20 

38.3 

15 

1 

50.2 

7 

45.0 

13 

39.8 

19 

43.0 

For  other  years  than  1899  the  following  quantities  should  be 
subtracted  or  added  to  the  tabular  values  : 


m. 


For  1895, 

subtract 

2.1 

For  1898, 

before  March  1, 

subtract  0.6 

For  1896, 

after  March  1, 

subtract  4.4 

For  1897, 

subtract 

3.1 

For  1898, 

subtract 

1.6 

For  1900, 

add 

1.1 

For  1901, 

add 

2.5 

For  1902, 

add 

3.8 

For  1903, 

add 

5.2 

For  1904, 

before  March  1, 

add 

6.6 

For  1904, 

after  March  1, 

add 

2.8 

For  1905, 

add 

4.1 

For  1906, 

add 

5.5 

For  1907, 

add 

6.8 

The  time  in  Table  V is  local  mean  astronomical  time, 
which  is  counted  from  noon  and  from  0 to  24  hours.  If  the 


TABLE  V.  POLARIS. 


155 


observer  lias  a watch  which  keeps  accurate  standard  time, 
he  can  reduce  the  astronomical  time  to  standard  time  by 
adding  or  subtracting  4 minutes  for  each  degree  of  longitude 
west  or  east  of  the  meridian  of  the  standard  time.  For  example, 
to  an  observer  in  longitude  90°  00'  the  east  elongation  of  Polaris 
will  occur  on  May  15,  1899,  at  3 : 50.3  a.m.  central  standard  time, 
but  to  an  observer  in  longitude  86°  36'  it  will  occur  13.6  minutes 
earlier,  or  at  3 : 36.7  a.m.  central  standard  time. 

To  obtain  the  times  for  any  calendar  day  other  than  the  first 
or  fifteenth  of  the  month,  subtract  3.94  minutes  for  every  day 
between  it  and  the  preceding  tabular  values,  or  add  3.94  min- 
utes for  each  day  between  it  and  the  following  tabular  value. 
For  example,  the  upper  culmination  of  Polaris  for  Nov.  10, 
1899,  occurs  at  10  hours  03.0  minutes  local  astronomical  time. 

Table  V is  computed  for  the  longitude  6 hours  west  of 
Greenwich  and  for  40  degrees  north  latitude.  To  correct  it 
for  other  longitudes  add  or  subtract  0.16  minutes  for  each 
hour  east  or  west  of  the  six-hour  meridian.  To  correct  the 
times  of  elongation  for  other  latitudes  add  or  subtract  0.13  min- 
utes to  the  times  of  ’west  or  east  elongations  for  each  degree 
south  of  40  degrees,  and  subtract  or  add  0.18  minutes  to  the 
times  of  west  or  east  elongation  for  each  degree  north  of  40 
degrees. 

As  an  example,  an  observer  in  north  latitude  42°  06'  and  west 
longitude  78°  45'  wishes  to  find  the  time  of  western  elongation 
of  Polaris  for  Sept.  27,  1896.  From  the  table  the  time  18h  50. 7m 
is  found  for  Sept.  27,  1899.  To  reduce  it  to  1896  the  correction 
4.4m  is  subtracted,  giving  18h  46. 3m.  The  correction  for  longi- 
tude is  0.1 2m  subtractive,  and  that  for  latitude  is  0.38m  additive. 
Thus  the  elongation  will  occur  at  18h  46. 6m  in  local  mean  as- 
tronomical time,  or  for  the  station  of  the  observer  at  7:01.6  A.M. 
eastern  standard  time.  The  probable  uncertainty  of  this  result 
is  about  0.7  minutes. 

Table  V has  been  compiled  from  information  kindly  fur- 
nished by  the  Superintendent  of  the  U.  S.  Coast  and  Geodetic 
Survey.  In  the  report  of  this  survey  for  1891,  Part  II,  page  8, 
a similar  table  for  1889  is  given. 


156 


TABLE  VI.  POLARIS, 


Table  VI. 

AZIMUTHS  OF  POLARIS  AT  ELONGATION  ON  JANUARY  1. 


Lat. 

1895 

1896 

1897 

1898 

1899 

1900 

1901 

1902 

25° 

1®  22'. 9 

1°  22' 

.6 

1°  22' 

.2 

1°  21'. 9 

1°  21 '.5 

1°  21'. 2 

1°  20'. 8 1°  20'. 5 

20 

23  .6 

23 

o 

22 

.9 

22  .5 

22  .2 

21  8 

21  .5 

21  .1 

27 

24  .3 

24 

.0 

23 

.6 

23  .3 

22  .9 

22  .5 

22  .2 

21  .9 

28 

25  .1 

24 

.7 

24 

.4 

24  .0 

23  .7 

23  .3 

23  .0 

22  .6 

29 

25  .9 

25 

.5 

25 

.2 

24  .8 

24  .5 

24  .1 

23  .8 

23  .4 

30 

26  .8 

26 

.4 

26 

.0 

25  .7 

25  .3 

24  .9 

24  .6 

24  .2 

31 

27  .6 

27 

.3 

26 

.9 

26  .5 

26  .2 

25  .8 

25  .5 

25  .1 

32 

28  .6 

28 

.2 

27 

.8 

27  .5 

27  .1 

26  .7 

26  .4 

26  .0 

33 

29  .6 

29 

.2 

28 

.8 

28  .5 

28  .1 

27  .7 

27  3 

2?  .0 

34 

30  .6 

30 

.3 

29 

.9 

29  .5 

29  .1 

28  .7 

28  .4 

28  .0 

35 

31  .7 

31 

.3 

31 

.0 

30  .6 

30  .2 

29  .8 

29  .4 

29  .0 

36 

32  .9 

32 

.5 

32 

.1 

31  .7 

31  .3 

30  .9 

30  .5 

30  .1 

37 

34  .1 

33 

.7 

33 

.3 

32  .9 

32  .5 

32  .1 

31  .7 

31  .3 

38 

35  .3 

34 

.9 

34 

.5 

34  .1 

33  .7 

33  .3 

33  .0 

32  .6 

39 

36  .7 

36 

.3 

35 

.9 

35  .5 

35  .1 

34  .7 

34  .3 

33  .9 

40 

38  .1 

37 

.7 

37 

.3 

36  .8 

36  .4 

36  .0 

35  .6 

35  .7 

41 

39  .6 

39 

.2 

38 

.8 

38  ,3 

37  .9 

37  .5 

37  .1 

36  .2 

42 

41  .1 

40 

.7 

40 

.3 

. 39  .8 

39  .4 

39  .0 

38  .6 

38  .2 

43 

42  .7 

42 

.3 

41 

.9 

41  .5 

41  .0 

40  .6 

40  .2 

39  .8 

44 

44  .4 

44 

.0 

43 

.6 

43  .1 

42  .7 

42  .3 

41  .8 

41  .4 

45 

46  .2 

45 

.8 

45 

.4 

44  .9 

44  .5 

44  .0 

43  .6 

43  .9 

46 

48  .2 

47 

.7 

47 

.3 

46  .8 

46  .4 

45  .9 

45  .5 

45  .2 

47 

50  .2 

49 

.7 

49 

.3 

48  .8 

48  .3 

47  .9 

47  .4 

46  .0 

48 

52  .3 

51 

.9 

51 

.4 

50  .9 

50  .4 

49  .9 

49  .5 

49  .0 

49 

54  .5 

54 

.1 

53 

.6 

53  .1 

52  .6 

52  .1 

51  .7 

51  .2 

50 

1°  56'. 9 

1°  56' 

.4 

1°  55' 

.9 

1°  55'. 4 

1°  54'. 9 

1°  54'. 5 

1°  54'. 0 

1°  537 

When  the  azimuth  is  required  with  a precision  less  than  one 
minute,  a correction  taken  from  the  following  supplementary 
table  should  be  applied.  For  example,  the  azimuth  of  Polaris 
for  latitude  43  degrees  on  Dec.  1,  1900,  is  1°  39'.  5. 


For 

Middle 

of 

Lat. 

25°. 

Lat. 

40°. 

Lat. 

50°. 

Jan 

-O'. 3 

-O'. 4 

-O'.  4 

Feb 

-0  .3 

-0  .3 

-0  .4 

Mar 

-0  .1 

-0  .2 

-0  .2 

Apr 

0 .0 

0 .0 

0 .0 

May 

-f-0  .2 

+0  .2 

4-0  v 2 

June 

+0  .2 

+0  .3 

40  .3 

For 

Middle 

of 

Lat. 

25°. 

Lat. 

40°. 

Lat. 

50°. 

July... 

+0'.2 

4-0'.  3 

-{-O'.  3 

Aug: 

+0  .1 

+0  .1 

-1-0  2 

Sept 

0 .0 

-0  .1 

-0  .1 

Oct 

-0  .2 

-0  .3 

-0  .3 

Nov 

-0  .5 

-0  .6 

-0  .7 

Dec 

-0  .6 

-0  .8 

-0  .9 

TABLE  VI.  POLARIS- 


157 


Table  VI. 

AZIMUTHS  OF  POLARIS  AT  ELONGATION  ON  JANUARY  1. 


Lat. 

1903 

1904 

1905 

1906 

1907 

1908 

1909 

1910 

25° 

1°  20' 

.1 

1°  19'. 8 

1°  19' 

.4 

1°  19'.  1 

1°  18'.7 

1°  18'. 4 

1°  18' 

M 

1°  17 '.7 

26 

20 

.8 

20  .5 

20 

.1 

19  .8 

19 

.4 

19  .1 

18 

.7 

18  .4 

27 

21 

.5 

21  .2 

20 

.8 

20  .5 

20 

.1 

19  .8 

19 

.4 

19  .1 

28 

22 

.2 

21  .9 

21 

.6 

21  .3 

20 

.9 

20  .5 

20 

.1 

19  .8 

29 

23 

.0 

22  .7 

22 

.4 

22  .1 

21 

. 7 

21  .3 

20 

.9 

20  .5 

30 

23 

.9 

23  .5 

23 

.1 

22  .8 

22 

.4 

22  .1 

21 

.7 

1 21  3 

31 

24 

.7 

24  .4 

24 

.0 

23  .6 

23 

.2 

22  .9 

22 

!5 

22  .2 

32 

25 

.6 

25  .3 

24 

.9 

24  .5 

24 

.1 

23  .8 

23 

.4 

1 23  .1 

33 

26 

.6 

26  .2 

25 

.9 

25  .5 

25 

.1 

24  .7 

24 

.3 

24  .0 

34 

27 

.6 

27  .2 

26 

.9 

26  .5 

26 

.1 

25  .7 

25 

.3 

25  .0 

35 

28 

.7 

28  .3 

27 

.9 

27  .5 

27 

.1 

26  .8 

26 

.4 

26  .0 

36 

29 

.8 

29  A 

29 

.0 

28  .6 

28 

.2 

27  .9 

27 

.5 

27  .1 

37 

30 

.9 

30  5 

30 

.1 

29  .7 

29 

.3 

29  .0 

28 

•6; 

28  .2 

38 

32 

.2 

31  .8 

31 

.4 

31  .0 

30 

.6 

30  .2 

29 

.8 

29  .4 

39 

33 

.5 

33  .1 

32 

.7 

32  .3 

31 

.8 

31  .4 

31 

•°| 

30  .6 

40 

34 

.8 

34  .4 

34 

.0 

33  .6 

33 

.2 

32  .8 

32 

.4 

32  .0 

41 

36 

.2 

35  .8 

35 

.4 

35  .0 

34 

.6 

34  .2 

33 

.8 

33  .4 

42 

37 

.7 

37  .3 

36 

.9 

36  .5 

36 

.0 

35  .6 

35 

.2 

34  .8 

43 

39 

.3 

38  .9 

38 

.5 

38  .1 

37 

.6 

37  .2 

36 

.8 

36  . 3 

44 

41 

.0 

40  .5 

40 

.1 

39  .7 

39 

.2 

38  .8 

38 

.4 

37.9 

45 

42 

. ( 

42  .3 

41 

.8 

41  .4 

40 

.9 

40  .5 

40 

.1! 

39  .6 

46 

44 

.6 

44  .2 

43 

.7 

43  .2 

42 

.7 

^2  .3 

41 

.9 

41  .4 

47 

46 

.5 

46  .0 

45 

.6 

45  .1 

44 

.6 

44  .2 

43 

.7 

43  .3 

48 

48 

.6 

48  .1 

47 

.7 

47'. 2 

46 

.7 

46  .3 

45 

.8 

45  .3 

49 

50 

.7 

50  .2 

49 

.8 

49  .3 

48 

.8 

48  .4 

47 

.9 

47  .4 

50 

1°  53' 

.0 

1°  52'. 5 

1°  52' 

.0 

1°  5P.5 

1°  5P 

.0 

1°  50'.  6 

1°  50' 

.1 

1°  49/.Oj 

The  azimuths  in  Table  VI  are  astronomical  azimuths  ; that  is, 
they  are  reckoned  from  the  true  north  toward  the  east  in  the 
case  of  east  elongation,  and  toward  the  west  in  the  case  of  west 
elongation.  For  intermediate  dates  and  latitudes  values  may  be 
found  by  interpolation.  Thus  for  latitude  42|  degrees  on 
March  10,  1896,  the  azimuth  is  1°  41. 4. 

The  above  table  is  taken  from  the  Report  of  the  U.  S.  Coast 
and  Geodetic  Survey  for  1891,  Part  II,  page  10.  An  azimuth 
deduced  by  the  help  of  the  auxiliary  correction  may  generally 
be  depended  upon  with  no  greater  error  than  O' .2. 


158 


TABLE  VII.  LINEAR  MEASURES, 


CONVERSION  OF  ENGLISH  INCHES  INTO  CENTIMETRES. 


Ins. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

Cm. 

Cm. 

Cm. 

Cm. 

Cm. 

Cm. 

Cm. 

Cm. 

Cm. 

Cm. 

0 

0.000 

2.540 

5.080 

7.620 

10.16 

12.70 

15.24 

17.78 

20.32 

22.88 

10 

25.40 

27.94 

30.48 

33.02 

35.56 

38.10 

40.64 

43.18 

45.72 

48.26 

20 

50.80 

53.34 

55.88 

58.42 

60.96 

63.50 

66.04 

68.58 

71.12 

73.66 

30 

76.20 

78.74 

81.28 

83.82 

86.36 

88.90 

91.44 

93.98 

96.52 

99.06 

40 

101.60 

104.14 

106.68 

109.22 

111.76 

114.30 

116.84 

119.38 

121.92 

124.48 

50 

127.00 

129.54 

132.08 

134.62 

137.16 

139.70 

142.24 

144.78 

147.32 

149.86 

00 

152.40 

154.94 

157.48 

160.02 

162.56 

165.10 

167.64 

170.18 

172.72 

175  26 

70 

177.80 

180.34 

182.88 

185.42 

187.96 

190.50 

193.04 

195.58 

198.12 

200.96 

80 

203.20 

205.74 

208.28 

210.82 

213.36 

215.90 

218.44 

220.98 

223.52 

226.06 

90 

228.60 

231.14 

233.68 

236.22 

238.76 

241.30 

243.84 

246 . 38 

248.92 

251.46 

100 

254.00 

256.54 

259.08 

261.62 

264.16 

266.70 

269  24 

271.78 

274.32 

276.80 

CONVERSION 

OF  CENTIMETRES 

INTO  ENGLISH  INCHES. 

Cm. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

Ins. 

Ins. 

Ins. 

Ins. 

Ins. 

Ins. 

Ins. 

Ins. 

Ins. 

Ins. 

0 

0.000 

0.394 

0.787 

1.181 

1.575 

1.969 

2.362 

2.756 

3.150 

3.543 

10 

3.937 

4.331 

4.742 

5.118 

5.512 

5.906 

i 6.299 

6.693 

7.087 

7.480 

20 

7.874 

8.268 

8.662 

9.055 

9.449 

9.843 

! 10. 236 

10.630 

11.024 

31.418 

30 

11.811 

12.205 

12.599 

12.992 

13.386 

13.780 

,14.173 

14.567 

14.961 

1.5.355 

40 

15.748 

16.142 

16.536 

16.929 

17.323 

17.717 

18.111 

18.504 

18.898 

j 19. 292 

50 

19.685 

20.079 

20.473 

20.867 

21.260 

21.654 

22.048 

22.441 

22.835 

23.229 

60 

23.622 

24.016 

24.410 

24.804 

25.197 

25.591 

25.985 

26.378 

26  772 

127.166 

70 

27.560 

27.953 

28.347 

28.741 

29.134 

29.528 

29.922 

30.316 

30.709 

1 31. 103 

80 

31.497 

31.890 

32.284 

32.678 

33.071 

33.465 

33.859 

34.253 

34.646 

'35.040 

90 

35.434 

35.827 

36.221 

36.615 

37.009 

37.402 

37.796 

38.190 

38.583 

138.977 

100 

39.370 

| 39.764 

40.158 

40.552 

40.945 

41 .339141 .733 

42.126 

42.520 

! 42. 914 

CONVERSION  OF  ENGLISH  FEET  INTO  METRES. 

Feet. 

0 

1 

2 

3 

4 

5 

6 

7. 

8 

9 

Met. 

Met. 

Met. 

Met. 

Met. 

Met. 

Met. 

1 Met. 

Met 

Met. 

0 

0.000 

0.3048 

0.6096 

0.9144 

1.2192 

1.5239 

1.8287 

12.1335 

2.4383 

2.7431 

10 

3.0479 

3.3527 

3.6575 

3.9623 

4.2671 

4.5719 

4.8767 

15.1815 

5.4863 

5.7913 

20 

6.0359 

6.4006 

6.7055 

7.0102 

7.3150 

7.6198 

7.9246 

; 8. 2294 

8.5342 

8.8390 

30 

9.1438 

9.4486 

9.7534 

10.058 

10.363 

10.668 

10.972 

11.277 

11.582 

11.887 

40 

12.192 

12.496 

12.801 

13.106 

13.411 

13.716 

14.020 

14.325 

14.630 

14.935 

50 

15.239 

15.544 

15.849 

16.154 

16.459 

16.763 

17.068 

17.373 

17.678 

17.983 

60 

18.287 

18.592 

18.897 

19.202 

19.507 

19.811 

20.116 

20.421 

20.726 

21.031 

70 

21.335 

21.640 

21.945 

22.250 

22.555 

22.859 

23.164 

23.469 

23  774 

24.079 

80 

24.383 

24.688 

24.993 

25.298 

25.602 

25.907 

26.212 

26.517 

26.822 

27.126 

90 

27.431 

27.736 

28.041 

28.346 

28.651 

28.955 

29.260 

29.565 

29.870 

30.174 

100 

30.479 

30.784 

I 31.089 

31.394 

31.698 

32.003 

32.308 

32.613 

32.918 

33.222 

CONVERSION  OF  METRES  INTO  ENGLISH  FEET. 

Met. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

Feet. 

Feet. 

Feet. 

Feet. 

Feet. 

Feet. 

Feet. 

Feet. 

Feet. 

Feet. 

0 

0.000 

3.2809 

6.5618 

9.8427 

13.123 

16.404 

19.685 

22.966 

26.247 

29.528 

10 

32.809 

36.090 

39.371 

42.651 

45.932 

49.213 

52.494 

55 .775 

59.056 

62.337 

20 

65.618 

68.899 

72.179 

75.461 

78.741 

82.022 

85.303 

88.584 

91.865 

95.146 

30 

98.427 

■ 101.71 

104.99 

108.27 

111.55 

114.83 

118.11 

121.39 

124.67 

127.96 

40 

131.24 

134.52 

137.80 

141.08 

144.36 

147.64 

150.92 

154.20 

157.48 

3 60. 76 

50 

164  04 

167.33 

170  61 

173.89 

177.17 

180.45 

183.73 

187.01 

190.29 

193.57 

60 

196.85 

200.13 

203.42 

206.70 

209.98 

213.26 

216.54 

219.82 

223.10 

226.38 

70 

229.66 

232.94 

236.22 

239.51 

242.79 

246.07 

249.35 

252.63 

255.91 

259.19 

80 

262.47 

265.75 

269.03 

272.31 

275.60 

278.88 

282.16 

285.44 

288.72 

292.00 

90 

295.28 

298.56 

391.84 

305.12 

308.40 

311.69 

314.97 

318.25 

321.53 

324.81 

100 

328.09 

331.37 

334.65 

! 337.93 

341.21 

344.491347.78 

351.06 

354.34 

357.62 

TABLE  VII.  LINEAR  MEASURES, 


159 


CONVERSION  OF 

ENGLISH  STATUTE-MILES  INTO  KILOMETRES. 

Miles. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

Kilo. 

Kilo. 

Kilo. 

Kilo. 

Kilo. 

Kilo. 

Kilo. 

Kilo. 

Kilo. 

Kilo. 

0 

0.0000  1.6093  3.2186 

4.8279 

6.4372 

8.0465 

9.6558 

11.2652 

12.8745 

14.4818 

10 

16.093 

17.702  19.312 

20.921 

22.530 

24.139 

25.749 

27.358 

28.967 

30.577 

20 

32.186 

33.795  35.405 

37.014 

38.623 

40.232 

41.842 

43.4511 

45.060 

46.670 

30 

48.279 

49.888  51.498 

53.107 

54.716 

56.325 

57.935 

59.544 

61.153 

62.763 

40 

64.372 

65.981 

67.591 

69.200 

70.809 

72.418 

74.028 

75.637 

77.246 

78.856 

50 

80.465 

82.074 

83.684 

85.293 

86.902 

88.511 

90.121 

91  730 

•93.339 

94.949 

60 

96.558 

98.167 

99  777 

101.39 

102.99 

104.60 

106  21 

107.82 

109.43 

111.04 

70 

112.65 

114.26 

115.87 

117.48 

119.08 

120.69 

122.30 

123.91 

125.52 

127.13 

80 

128.74 

130.35 

131.96 

133.57 

135.17 

136.78 

138.39 

140.00 

141.61 

143.22 

90 

144.85 

146.44 

148.05 

149.66 

151.26 

152.87 

154.48 

156.09 

157.70 

159.31 

100 

160.93 

162.53 

164  14 

165  75 

167.35 

168.96 

170.57 

172.18 

173.79 

175.40  ) 

CONVERSION  OF 

KILOMETRES  INTO  ENGLISH 

: STATUTE-MILES 

Kilom. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

Miles. 

Miles. 

Miles. 

Miles. 

Miles. 

Miles. 

Miles. 

Miles. 

Miles. 

Miles. 

0 

0.0000 

0.6214 

1 2427 

1.8641 

2.4855 

3.1069 

3.7282 

4.3497 

4.9711 

5.5924 

10 

6.2138 

6.8352 

7.4565 

8.0780 

8.6994 

9.3208 

9.9421 

10.562 

11.185 

11.805 

20 

12.427 

13.049 

13.670 

14.292 

14.913 

15.534 

16.156 

16.776 

17.399 

18.019 

30 

18.641 

19.263 

19.884 

20.506 

21.127 

21.748 

23  370 

22.990 

23.613 

24.233 

40 

24.855 

25.477 

26.098 

26.720 

27.341 

27.962 

28.584 

29.204 

29.827 

30.447 

50 

31.069 

31.690 

32.311 

32.933 

33.554 

34.175 

34.797 

35.417 

36.040 

36.660 

60 

37.282 

37.904 

38.525 

39.147 

39.768 

40.389 

41.011 

41.631 

42.254 

42.874 

70 

43.497 

44.118 

44.739 

45.361 

45.982 

46.603 

47.225 

47.845 

48.468 

49.088 

80 

49.711 

50.332 

50.953 

51.575 

52.196 

52.817 

53.439 

54.059 

54.682 

: 55.302 

90 

55.924 

56 . 545 

57.166 

57.788 

58.409 

59.030 

59.652 

60.272 

60.895 

61.515 

100 

62.138 

62.759 

63.380 

64.002 

64.623 

65.244 

65.866 

66.486 

67.109 

i 67.729 

TABLE 

VIII. 

LENGTH  IN  FEET  OF 

1'  ARCS  OF  LATITUDE 

AND  LONGITUDE. 

Lat. 

V Lat. 

V Long. 

Lat. 

V Lat. 

V Long. 

1° 

6045 

6085 

31° 

6061 

5222 

2° 

6045 

6083 

32° 

6062 

5166 

3° 

6045 

6078 

33° 

6063 

5109 

4° 

6045 

6071 

34° 

6064 

5051 

5° 

6045 

6063 

35° 

6065 

4991 

6° 

6045 

6053 

36° 

6066 

4930 

7° 

6016 

6041 

37° 

6067 

4867 

8° 

6046 

6027 

38° 

6068 

4802 

9° 

6046 

6012 

39° 

6070 

4736 

10° 

6047 

• 5994 

40° 

6071 

4669 

11° 

6047 

5975 

41° 

6072 

4600 

12° 

, 6048 

5954 

42° 

6073 

4530 

13° 

6048 

5931 

43° 

6074 

4458 

14° 

6019 

5907 

44° 

6075 

4385 

15° 

6019 

5880 

45° 

6076 

4311 

16° 

6050 

5852 

46° 

6077 

4235 

17° 

6050 

5822 

47° 

6078 

4158 

18° 

6051 

5790 

. 48° 

6079 

4080 

19° 

6052 

5757 

49° 

6080 

4001 

20° 

6052  • 

5721 

50° 

6081 

3920 

21° 

6053 

5684 

51° 

6082 

3838 

22° 

6(54 

5646 

52° 

6084 

3755 

23° 

6054 

5605 

53° 

6085 

3671 

24° 

6055 

5563 

54° 

6086 

3586 

25° 

6056 

5519 

55° 

6087 

3199 

26° 

6057 

5474 

56° 

6088 

3413 

27° 

6058 

5427 

57° 

6089 

3323 

28° 

6059 

5378 

£8° 

6090 

3233 

29° 

6060 

5327 

59° 

6091 

3142 

30° 

6061 

5275 

60° 

6092 

3051 

160 


TABLE  IX.  INCLINED  DISTANCES. 


Table  IX. 

REDUCTION  OF  INCLINED  DISTANCES  TO  THE  HORIZONTAL. 
Inclined  Distance  = 100  feet. 


Slope. 

Correction 

Horizontal 

Distance. 

0° 

(XT 

100.000 

30 

0.004 

99.996 

1 

00 

0.015 

99.985 

30 

0.034 

99.966 

2 

00 

0.061 

99.939 

30 

0.095 

99.905 

3 

00 

0.137 

99.863 

30 

0.187 

99.813 

4 

00 

0.244 

99.756 

30 

0.308 

99.692 

5 

00 

0.381 

99.619 

30 

0.460 

99.540 

6 

00 

0.548 

99.452 

30 

0.643 

99  357 

7 

00 

0.745 

99.255 

30 

0.856 

99.144 

Slope. 

Correction. 

Horizontal 

Distance. 

8°  00' 

0.973 

99.027 

30 

1.098 

98.902 

9 00 

1.231 

98.769 

30 

1.371 

98.629 

10  00 

1.519 

98.481 

30 

1.675 

98.325 

11  00 

1.837 

98.163 

30 

2.008 

97.992 

12  00 

2.185 

97.814 

30 

2.370 

97.630 

13  00 

2.563 

97.437 

30 

2.763 

97.237 

14  00 

2.970 

97.030 

30 

3.185 

96.815 

15  00 

3 407 

96.593 

30 

3.637 

96.363 

ANSWERS  TO  PROBLEMS. 

Prob.  1 : A = 10°  14',  B — 7°  82'.  Prob.  2 : azimuth  of  BE 
= 106°  45'.  Prob.  3 : latitude  = + 2458.2  feet,  longitude 
— _|_  5379.4  feet.  Prob.  4 : area  — 5 acres,  104  rods,  84  square 
feet.  Prob.  5 : for  BC,  + 382.1  feet,  and  -f-  823.3  feet.  Prob. 
6 : Area  = 11  acres,  41  rods,  203  square  feet.  Prob.  8 : dis- 
tance = 10340  feet.  Prob.  9:  M is  226.6  feet  above  N.  Prob. 
10  : AOD  — 117°  52£#,  COB  = 22°  01£'.  Prob.  11  : true  area 
= 7 acres,  146  rods,  222  square  feet.  Prob.  13 : maximum 
declination  8°  03'  in  January,  1916.  Prob.  14  : area  = 3 acres,  0 
roods,  4.7  square" rods.  Prob.  18  : N 78°  06  W,  26  links,  for  A ; 
S 74°  35'  W,  56  links  for  C.  Prob.  20  : 476.954  and  477.715 
chains.  Prob.  23  : error  = 0.025  feet.  Prob.  28  : pull  = 14.9 
pounds.  Prob.  30 : latitude  = 2000.000  feet,  longitude  = 
4000.000  feet.  Prob.  31:  83£  feet,  398.6  acres.  Prob.  34: 
902.6  and  417.1  for  the  first  point. 


Table  X. 


REDUCTION  OF  STADIA  READINGS 

TO 

HORIZONTAL  DISTANCES 

AND  TO 

DIFFERENCES  OF  ELEVATION. 


This  table  was  computed  by  Professor  Arthur  Winslow, 
State  Geologist  of  Missouri. 


162 


TABLE  X.  STADIA  REDUCTIONS, 


Table  X. 

STADIA  REDUCTIONS  FOR  READING  100. 


Minutes. 

0 

1 

° 

2 

3 

D 

Hor. 

Diff. 

Hor. 

Diff. 

Hor. 

Diff. 

Hor. 

Diff. 

Dist. 

Elev. 

Dist. 

Elev. 

Dist. 

Elev. 

Dist. 

Elev. 

O' 

100.00 

.00 

99.97 

1.74 

99.88 

3.49 

99.73 

5.23 

2 

“ 

.06 

“ 

1.80 

99.87 

3.55 

99.72 

5.28 

4 

u 

.12 

“ 

1.86 

“ 

3.60 

99.71 

5.34 

6 

u 

.17 

99.96 

1.92 

“ 

3.66 

“ 

5.40 

8 

tt 

.23 

“ 

1.98 

99.86 

3.72 

99.70 

5.46 

10 

tt 

.29 

“ 

2.04 

3.78 

99.69 

5.52 

12 

« 

.35 

tt 

2.09 

99.85 

3.84 

U 

5.57 

14 

tt 

.41 

99.95 

2.15 

“ 

3.90 

99.68 

5.63 

16 

.47 

2.21 

99.84 

3.95 

tt 

5.69 

18 

“ 

.52 

It 

2.27 

“ 

4.01 

99.67 

5.75 

20 

“ 

.58 

2.33 

99.83 

4.07 

99.66 

5.80 

22 

<t 

.64 

99.94 

2.38 

4.13 

5.86 

24 

it 

.70 

“ 

2.44 

99.82 

4.18 

99.65 

5.92 

26 

99.99 

.76 

tt 

2.50 

“ 

4.24 

99.64 

5.98 

28 

“ 

.81 

99.93 

2.56 

99.81 

4.30 

99.63 

6.04 

30 

tt 

.87 

44 

2.62 

u 

4.36 

“ 

6.09 

32 

tt 

.93 

tt 

2.67 

99.80 

4.42 

99.62 

6.15 

34 

(t 

.99 

44 

2.73 

“ 

4.48 

“ 

6.21 

36 

1.05 

99.92 

2.79 

99.79 

4.53 

99.61 

6.27 

38 

“ 

1.11 

“ 

2.85 

“ 

4.59 

99.60 

6.33 

40 

<( 

1.16 

“ 

2.91 

99.78 

4.65 

99.59 

6.38 

42 

t* 

1.22 

99.91 

2.97 

4.71 

6.44 

44 

99.98 

1.28 

“ 

3.02 

99.77 

4.76 

99.58 

6.50 

46 

“ 

1.34 

99.90 

3 08 

“ 

4.82 

99.57 

6.56 

48 

“ 

1.40 

“ 

3.14 

99.76 

4.88 

99.56 

6.61 

50 

“ 

1.45 

“ 

3.20 

44 

4.94 

“ 

6.67 

52 

1.51 

99.89 

3.26 

99.75 

4.99 

99.55 

6.73 

54 

“ 

1.57 

“ 

3.31 

99.74 

5.05 

99.54 

6.78 

56 

99.97 

1.63 

tt 

3.37 

“ 

5.11 

99.53 

6.84 

58 

tt 

1.69 

99.88 

3.43 

99.73 

5.17 

99.52 

6.90 

60 

tt 

1.74 

‘k 

3.49 

tt 

5.23 

99  51 

6.96 

c+/=  .75 

.75 

.01 

.75 

.02 

.75 

.03 

.75 

.05 

c+f=  1.00 

1.00 

.01 

1.00 

.03 

1.00 

.04 

1.00 

.06 

C+f=  1.25 

1.25 

.02 

1.25 

.03 

1.25 

.05 

1.25 

.08  1 

TABLE  X.  STADIA  REDUCTIONS, 


1G3 


Table  X. 

STADIA  REDUCTIONS  FOR  READING  100. 


Minutes. 

4 

O 

5 

o 

6 

O 

7 

O 

Hor. 

Diff. 

Hor. 

Diff. 

Hor. 

Diff. 

Hor. 

Diff. 

i 

Dist. 

Elev. 

Dist. 

Elev. 

Dist. 

Elev. 

Dist. 

Elev. 

O' 

99.51 

6.96 

99.24 

8.68 

98.91 

10.40 

98.51 

12.10 

2 

“ 

7.02 

99.23 

8.74 

98.90 

10.45 

98.50 

12.15 

4 

99.50 

7.07 

99.22 

8.80 

68.88 

10.51 

98.48 

12  21 

6 

99.49 

7.13 

99.21 

8.85 

98.87 

10.57 

S8.47 

12.26 

8 

99.48 

7.19 

99.20 

8.91 

98.86 

10  62 

98.46 

12.32 

10 

99.47 

7.25 

99.19 

8.97 

98.85 

10.68 

98.44 

12.38 

12 

99.46 

7.30 

99.18 

9.03 

98.83 

10.74 

98.43 

12.43 

14 

“ 

7.36 

99.17 

9.08 

98.82 

10.79 

98.41 

12.49 

16 

99.45 

7.42 

99.16 

9.14 

98.81 

10.85 

98.40 

12.55 

18 

99.44 

7.48 

99.15 

9.20 

98.80 

10.91 

98.39* 

12.60 

20 

99.43 

7.53 

99.14 

9.25 

98.78 

10.96 

98.37 

12.66 

22 

99.42 

7.59 

99.13 

9.31 

98.77 

11.02 

98.36 

12.72 

24 

99.41 

7.65 

99.11 

9.37 

98.76 

11.08 

98.34 

12.77 

26 

99.40 

7.71 

99.10 

9.43 

98.74 

11.13 

98.33 

12.83 

28 

99.39 

7.76 

99.09 

9.48 

98.73 

11.19 

98.31 

12.88 

30 

99.38 

7.82 

99.08 

9.54 

98.72 

11.25 

98.29 

12.94 

32 

99.38 

7.88 

99.07 

9.60 

98.71 

11.30 

98.28 

13.00 

34 

99.37 

7.94 

99.06 

9.65 

98  69 

11.36 

98.27 

13.05 

. 36 

99.36 

7.99 

99.05 

9.71 

98.68 

11.42 

98.25 

13.11 

38 

99.35 

8.05 

99.04 

9.77 

98.67 

11.47 

98.24 

13.17 

40 

99.34 

8.11 

99.03 

9.83 

98.65 

11.53 

98.22 

13.22 

42 

99.33 

8.17 

99.01 

9.88 

98.64 

11.59 

98.20 

13  28 

44 

99.32 

8.22 

99.00 

9.94 

98.63 

11.64 

98.19 

13.33 

46 

99.31 

8.28 

98.99 

10.00 

98.61 

11.70 

98.17 

13  39 

48 

99.30 

8.34 

98.98 

10.05 

98.60 

11.76 

98.16 

13  45 

50 

99.29 

8.40 

98.97 

10.11 

98.58 

11.81 

98.14 

13.50. 

52 

99.28 

8.45 

98.96 

10.17 

98.57 

11.87 

98.13 

13.56 

54 

99.27 

8.51 

98.94 

10.22 

98.56 

11.93 

98.11 

13.61 

56 

99.26 

8.57 

98.93 

10.28 

98.54 

11.98 

98.10 

13.67 

58 

99.25 

8.63 

98.92 

10.34 

98.53 

12.04 

98.08 

13.73 

60 

99.24 

8.68 

98.91 

10.40 

98.51 

12.10 

98.06 

13.78 

c+f-  .75 

.75 

.06 

.75 

.07 

.75 

.08 

.74 

.10 

c+f=  1.00 

1.00 

.08 

.99 

.09 

.99 

.11 

.99 

.13 

c+/=  1.25 

1.25 

.10 

1.24 

.11 

1.24 

.14 

1.24 

.16 

164 


TABLE  X.  STADIA  REDUCTIONS. 


Table  X. 

STADIA  REDUCTIONS  FOR  READING  100. 


Minutes. 

8 

O 

9 

O 

10° 

11° 

Hor. 

Dist. 

Diff. 

Elev. 

Hor. 

Dist. 

Diff. 

Elev. 

Hor. 

Dist. 

Diff. 

Elev. 

Hor. 

Dist. 

Diff. 

Elev. 

O' 

98.06 

13.78 

97.55 

15.45 

96.98 

17.10 

96.36 

18.73 

2 

98.05 

13.84 

97.53 

15.51 

96.96 

17.16 

96  34 

18.78 

4 

98.03 

13.89 

97.52 

15.56 

96.94 

17.21 

96.32 

18.84 

6 

98.01 

13.95 

97.50 

15.62 

96.92 

17.26 

96.29 

18.89 

8 

98.00 

14.01 

97.48 

15.67  - 

96.90 

17.32 

96.27 

18.95 

10 

97.98 

14.06 

97.46 

15.73 

96.88 

17.37 

96.25 

19.00 

12 

97.97 

14.12 

97.44 

15.78 

96.86 

17.43 

96.23 

19.05 

14 

97.95 

14.17 

97.43 

15.84 

96.84 

17.48 

96.21 

19.11 

16 

97.93 

14.23 

97.41 

15.89 

96.82 

17.54 

96.18 

19.16 

18 

97.92 

14.28 

97.39 

15.95 

96.80 

17.59 

96.16 

19.21 

20 

97.90 

14.34 

97.37 

16.00 

96.78 

17.65 

96.14 

19.27 

22 

97.88 

14.40 

97.35 

16.06 

96.76 

17.70 

96.12 

19.32 

24 

97.87 

14.45 

97.33 

16.11 

96.74 

17.76 

96.09 

19.38 

26 

97.85 

14.51 

97.31 

16.17 

96.72 

17.81 

96.07 

19.43 

28 

97.83 

14.56 

97.29 

16.22 

96.70 

17.86 

96.05 

19.48 

30 

97.82 

14.62 

97.28 

16.28 

96.68 

17.92 

96.03 

19.54 

32 

97.80 

14.67 

97.26 

16.33 

96.66 

17.97 

96.00 

19.59 

34 

97.78 

14.73 

97.24 

16.39 

96.64 

18.03 

95. 98 

19.64 

36 

97.76 

14.79 

97.22 

16.44 

96.62 

18.08 

95.96 

19.70 

38 

97.75 

14.84 

97.20 

18.50 

96.60 

18.14 

95.93 

19.75 

40 

97.73 

14.90 

97.18 

16.55 

96.57 

18.19 

95.91 

19.80 

42 

97.71 

14.95 

97.16 

16.61 

96.55 

18.24 

95.89 

19.86 

44 

97.69 

15.01 

97.14 

16.66 

96.53 

18.30 

95.86 

19.91 

46 

97.68 

15.06 

97.12 

16.72 

96.51 

18.35 

95.84 

19.96 

48 

97.66 

15.12 

97.10 

16.77 

96.49 

18.41 

95.82 

20.02 

50 

97.64 

15.17 

97.08 

16.83 

96.47 

18.46 

95.79 

20.07 

52 

97.62 

15.23 

97.06 

16.88 

96.45 

18.51 

95  >77 

20.12 

54 

97.61 

15.28 

97.04 

16.94 

96.42 

18.57 

95.75 

20.18 

56 

97.59 

15.34 

97.02 

16.99 

96.40 

18.62 

95.72 

20.23 

58 

97.57 

15.40 

97.00 

17.05 

96.38 

18.68 

95.70 

20.28 

60 

97.55 

15.45 

96.98 

17.10 

96.36 

18.73 

95.68 

20.34 

c +/  = 

.75 

.74 

.11 

.74 

.12 

.74 

.14 

.73 

.15 

c + / = 

V.00 

.99 

.15 

.99 

.16 

.98 

.18 

.98 

.20 

c+/  = 

1.25 

1.23 

.18 

1.23 

.21 

1.23 

.23 

1.22 

.25 

TABLE  X.  STADIA  liEDUCTlOXS. 


16o 


Table  X. 

STADIA  REDUCTIONS  FOR  READING  100. 


Minutes. 

12° 

13° 

14° 

15° 

Hor. 

Dist. 

Diff. 

Elev. 

Hor. 

Dist. 

Diff. 

Elev. 

Hor. 

Dist. 

Diff. 

Elev. 

Hor. 

Dis^t. 

Diff. 

Elev. 

O' 

95.68 

20.34 

94.94 

21.92 

94.15 

23.4' 

93.30 

25.00 

2 

95.65 

20.39 

94.91 

21.97 

94.12 

23.52 

93.27 

25.05 

4 

95.63 

20.44 

94.89 

22.02 

94.09 

23.58 

93.24 

25.10 

6 

95.61 

20.50 

94.86 

22.08 

94.07 

23.63 

93.21 

25.15 

8 

95-58 

20.55 

94.84 

22.13 

94.04 

23.68 

93.18 

25.20 

10 

95.56 

20.60 

94.81 

22.18 

94.01 

23.73 

93.16 

25 . 25 

•12 

95.53 

20.66 

94.79 

22.23 

93  98 

23.78 

93.13 

25.30 

14 

C5.51 

20.71 

94.76 

22.28 

93.95 

23.83 

93.10 

25.35 

16 

95.49 

20.76 

94.73 

22.34 

93.93 

23.88 

93.07 

25.40 

18 

95.46 

20.81 

94.71 

22.39 

93.90 

23.93 

93.04 

25.45 

20 

95.44 

20.87 

94.68 

22.44 

93.87 

23.99 

93.01 

25.50 

22 

95.41 

20.92 

94.66 

22.49 

93.84 

24.04 

92.98 

25.55 

24 

95.39 

20.97 

94.63 

22.54 

93.81 

24.09 

92.95 

25.60 

26 

95.36 

21.03 

94.60 

22.60 

93.79 

24.14 

92.92 

25.65 

* 28 

95.34 

21.08 

94.58 

22.65 

93.76 

24.19 

92.89 

25.70 

30 

95.32 

21.13 

94.55 

22.70 

93.73 

24.24 

92.86 

25.75 

32 

95.29 

21.18 

94.52 

22.75 

93.70 

24.29 

92  83 

25.80 

34 

95.27 

21.24 

94.50 

22.80 

93.67 

24.34 

92.80 

25  85 

36 

95.24 

21.29 

94.47 

22.85 

93.65 

24.39 

92.77 

25.90 

38 

95.22 

21  34 

94.44 

22.91 

93.62 

24.44 

92.74 

25.95 

40 

95.19 

21.39 

94.42 

22.96 

93.59 

24.49 

92.71 

26.00 

42 

95.17 

21.45 

94.39 

23.01 

93.56 

24.55 

92  68 

26.05 

44 

95.14 

21.50 

94.36 

23.06 

93.53 

24.60 

92.65 

26.10 

46 

95.12 

21.55 

94.34 

23.11 

93.50 

24.65 

92.62 

26.15 

48 

95.09 

21.60 

94.31 

23.16 

93.47 

24.70 

92.59 

26.20 

50 

95.07 

21.66 

94.28 

23.22 

93.45 

24.75 

92.56 

26.25 

52  , 

95.04 

21.71 

94.26 

23.27 

93.42 

24.80 

92.53 

26.30 

54 

95.02 

21.76 

94.23 

23.32 

93.39 

24.85 

92.49 

26.35 

56 

94.99 

21.81 

94.20 

23.37 

93.36 

24.90 

92.46 

26.40 

58 

94.97 

21.87 

94.17 

23.42 

93.33 

24.95 

92.43 

26.45 

60 

94.94 

21.92 

94.15 

23.47 

93.30 

25.00 

92.40 

26.50 

c4-/=  -75 

.73 

.16 

.73 

- .17 

.73 

.19 

.72 

.20 

c-j-/  = 1.00 

.98 

.22 

.97 

.23 

.97 

.25 

.96 

.27 

c+f  = 1.25 

1.22 

.27 

1.21 

.29 

1.21 

.31 

1.20 

.34 

166  TABLE  X.  STADIA  REDUCTIONS. 


Table  X. 

STADIA  REDUCTIONS  FOR  READING  100. 


Minutes. 

16° 

17° 

18° 

19° 

Hor. 

Dist. 

Diff. 

Elev. 

Hor. 

Dist. 

Diff. 

Elev. 

Hor. 

Dist. 

Diff. 

Elev. 

Hur. 

Dist. 

Diff. 

Elev. 

O' 

92.40 

26.50 

91.45 

27.96 

90.45 

29.39 

89.40 

30.78 

2 

92.37 

26.55 

91.42 

28.01 

90.42 

29.44 

89  36 

30.83 

4 

92.34 

26.59 

91.39 

28.06 

90.38 

29.48 

89.33 

30.87 

6 

92.31 

26.64 

91.35 

28.10 

90.35 

29.53 

89.29 

30.92 

8 

92.28 

26.69 

91.32 

28.15 

90.31 

29.58 

89.26 

30.97 

10 

92.25 

26.74 

91.29 

28.20 

90.28 

29.62 

89.22 

31.01 

12 

92.22 

26.79 

91.26 

28.25 

90.24 

29.67 

89.18 

31.06 

14 

92.19 

26.84 

91.22 

28.30 

90.21 

29.72 

89.15 

31.10 

16 

92.15 

26.89 

91.19 

28.34 

90.18 

29.76 

89.11 

31.15 

18 

92.12 

26.94 

91.16 

28.39 

90.14 

29.81 

89.08 

31.19 

20 

92.09 

26.99 

91.12 

28.44 

90.11 

29.86 

89.04 

31.24 

22 

92.06 

27.04 

91.09 

28.49 

90.07 

29.90 

89.00 

31.28 

24 

92.03 

27.09 

91.06 

28.54 

90.04 

29.95 

88.96 

31.33 

26 

92.00 

27.13 

91.02 

28.58 

90.00 

30.00 

88.93 

31.38. 

28 

91.97 

27.18 

90.99 

28.63 

89.97 

30.04 

88.89 

31.42 

30 

91.93 

27.23 

90.96 

28.68 

89.93 

30.09 

88.86 

31.47 

32 

91.90 

27.28 

90.92 

28.73 

89.90 

30.14 

88.82 

31.51 

34 

91.87 

27.33 

90.89 

28.77 

89.86 

30.19 

88.78 

31.56 

36 

91.84 

27.38 

90.86 

28.82 

89.83 

30.23 

88.75 

31.60 

38 

91.81 

27.43 

90.82 

28.87 

89.79 

30.28 

88.71 

31.65 

40 

91.77 

27.48 

90.79 

28.92 

89.76 

30.32 

88.67 

31.69 

42 

91.74 

27.52 

90.76 

28.96 

89.72 

30.37 

88.64 

31.74 

44 

91.71 

27.57 

90.72 

29.01 

89.69 

30.41 

88.60 

31.78 

46 

91.68 

27.62 

90.69 

29.06 

89.65 

30.46 

88.56 

31.83 

48 

91.65 

27.67 

90.66 

29.11 

89.61 

30.51 

88.53 

31.87 

50 

91.61 

27.72 

90.62 

29.15 

89.58 

30.55 

88.49 

31.92 

52 

91.58 

27.77 

90.59 

29.20 

89.54 

30.60 

88.45 

31.96 

54 

91.55 

27.81 

90.55 

29.25 

89.51 

30.65 

88.41 

32.01 

56 

91.52 

27.86 

90.52 

29.30 

89.47 

30.69 

88.38 

32.05 

58 

91.48 

27.91 

90.48 

29.34 

89.44 

30.74 

88.34 

32.09 

60 

91.45 

27.96 

90.45 

29.39 

89.40 

30.78 

88.30 

32.14 

c+/=  .75 

.72 

.21 

.72 

.23 

.71 

.24 

.71 

.25 

c+/=  1.00 

.96 

.28 

.95 

.30 

.95 

.32 

!94 

.33 

c+/=  1-25 

1.20 

.36 

1.19 

.38 

1.19 

•40  | 

1.18 

.42 

TABLE  X.  STADIA  REDUCTIONS, 


167 


Table  X. 

STADIA  REDUCTIONS  FOR  READING  100. 


Minutes. 

20° 

21° 

no 

iSD 

23° 

Hor. 

Dist. 

Dili. 

Elev. 

Hor. 

Dist. 

Diff, 

Elev. 

Hor. 

Dist. 

Diff. 

Elev. 

Hor. 

Dist. 

Diff. 

Elev. 

O' 

88.30 

32.14 

87.16 

33.46 

85.97 

34.73 

84.73 

35.97 

2 

88.26 

32.18 

87.12 

33.50 

85.93 

34.77 

84.69 

36.01 

4 

88.23 

32.23 

87.08. 

33.54 

85.89 

34.82 

84.65 

36.05 

6 

88.19 

32.27 

87.04 

33.59 

85.85 

34.86 

84.61 

36.09 

8 

88.15 

32.32 

87.00 

33.63 

85.80 

34  90 

84.57 

36.13 

10 

88.11 

32.36 

86.96 

33.67 

85.76 

34.94 

84.52 

36.17 

12 

88.08 

32.41 

86.92 

33.72 

85.72 

34.98 

84.48 

36.21 

14 

83.04 

32.45 

86.88 

33.76 

85.68 

35.02 

84.44 

36.25 

16 

88.00 

32.49 

86.84 

33.80 

85.64 

35.07 

84.40 

36.29 

18 

87.96 

32.54 

86.80 

33.84 

85.60 

35.11 

84.35 

36.33 

20 

87.93 

32.58 

86.77 

33.89 

85.56 

35.15 

84.31 

36.37 

22 

87.89 

32.63 

86.73 

33.93 

85.52 

35.19 

84.27 

36.41 

24 

87.85 

32.67 

86.69 

33.97 

85.48 

35.23 

84.23 

36.45 

26 

87.81 

32.72 

86.65 

34.01 

85.44 

35.27 

84.18 

36.49 

28 

87.77 

32.76 

86.61 

34.06 

85.40 

35.31 

84.14 

36.53 

30 

87.74 

32.80 

86.57 

34.10 

85.36 

35  36 

84.10 

36.57 

32 

87.70 

32.85 

86.53 

34.14 

85.31 

35.40 

84.06 

36.61 

34 

87.66 

32.89 

86.49 

34.18 

85.27 

35.44 

84.01 

36.65 

36 

87.62 

32.93 

86.45 

34.23 

85.23 

35.48 

83.97 

36.69 

38 

87.58 

32.98 

86.41 

34.27 

85.19 

35.52 

83.93 

36.73 

40 

87.54 

33.02 

86.37 

34.31 

85.15 

35.56 

83.89 

36.77 

42 

87.51 

33.07 

86.33 

34.35 

85.11 

35.60 

83.84 

36.80 

44 

87.47 

33.11 

86.29 

34.40 

85.07 

35.64 

83.80 

36.84 

46 

87.43 

33.15 

86 . 25 

34.44 

85.02 

35.68 

83.76 

36.88 

48 

87.39 

33.20 

86  21 

34.48 

84.98 

35.72 

83.72 

36.92 

50 

87.35 

33.24 

86.17 

34.52 

84.94 

35.76 

83.67 

36.96 

52 

87.31 

33.28 

86.13 

34.57 

84.90 

35.80 

83.63 

37.00 

54 

87.27 

33.33 

86.09 

34.61 

84.86 

35.85 

83.59 

37.01 

56 

87.24 

33.37 

86.05 

34.65 

84.82 

35.89 

83.51 

37.08 

58 

87.20 

33.41 

86.01 

34.69 

84.77 

35.93 

83.50 

37.12 

60 

87.16 

33.46 

85.97 

34.73 

84.73 

35.97 

83.46 

37.16 

c+f=  .75 

.70 

.26 

.70 

.27 

.69 

.29 

.69 

.30 

c+f=  1.00 

.94 

.35 

.93 

.37 

.92 

.38 

.92 

.40 

c+f  =1.25 

1.17 

.44 

1.16 

.46 

1.15 

.48 

1.15 

.50 

168 


TABLE  X.  STADIA  REDUCTIONS, 


Table  X. 

STADIA  REDUCTIONS  FOR  READING  100. 


Minutes. 

24° 

25° 

26° 

27° 

Hor. 

Dist. 

Diff. 

Elev. 

Hor. 

Dist 

Diff. 

Elev. 

Hor. 

Dist. 

Diff. 

Elev. 

Hor. 

Dist. 

Diff. 

Elev. 

0' 

83.46 

37.16 

82.14 

38.30 

80.78 

39.40 

79.39 

40.45 

2 

83.41 

37.20 

82.09 

38.34 

80.74 

39.44 

79.34 

40.49 

4 

83.37 

37.23 

82.05 

38.38 

80.69 

39.47 

79.30 

40.52 

6 

83.33 

37.27 

82.01 

38.41 

80.65 

39.51 

79.25 

40.55 

8 

83.28 

37.31 

81.96 

38.45 

80.60 

39.54 

79  20 

40.59 

10 

83.24 

37.35 

81.92 

38.49 

80.55 

39.58 

79.15 

40.62 

12 

83.20 

37.39 

81.87 

38.53 

80.51 

39.61 

79.11 

40.66 

14 

83.15 

37.43 

81.83 

38.56 

80.46 

39.65 

79.06 

40.69 

16 

83.11 

37.47 

81.78 

38  60 

80.41 

39.69 

79.01 

40.72 

18 

83.07 

37.51 

81.74 

38.64 

80.37 

39.72 

78.96 

40.76 

20 

83.02 

37.54 

81.69 

38.67 

80.32 

39.76 

78.92 

40.79 

22 

82.98 

37.58 

81.65 

38.71 

80.28 

39.79 

78.87 

40.82 

24 

82.93 

37.62 

81.60 

38.75 

80.23 

39.83 

78.82 

40.86 

26 

82.89 

37.66 

81.56 

38.78 

80.18 

39.86 

78.77 

40  89 

28 

82.85 

37.70 

81.51 

38.82 

80.14 

39.90 

78.73 

40.92 

30 

82.80 

37.74 

81.47 

38.86 

80.09 

39.93 

78.68 

40.96 

32 

82.76 

37.77 

81.42 

38.89 

80.04 

39.97 

78.63 

40.99 

34 

82.72 

37.81 

81.38 

38.93 

80.00 

40.00 

78.58 

41.02 

36 

82.67 

37.85 

81.33 

38.97 

79.95 

40.04 

78.54 

41.06 

38 

82.63 

37.89 

81.28 

39.00 

79.90 

40.07 

78.49 

41.09 

40 

82.58 

37.93 

81.24 

39.04 

79.86 

40.11 

78.44 

41.12 

42 

82.54 

37.96 

81.19 

39.08 

79.81 

40.14 

78.39 

41.16 

44 

82.49 

38.00 

81.15 

39.11 

79.76 

40.18 

78.34 

41.19 

46 

82.45 

38.04 

81.10 

39.15 

79.72 

40.21 

78.30 

41.22 

48 

82.41 

38.08 

81.06 

39.18 

79.67 

40.24 

78.25 

41.26 

50 

82.36 

38.11 

8k  01 

39.22 

79.62 

40.28 

78.20 

41.29 

52 

82.32 

38.15 

80.97 

39.26 

79.58 

40.31 

78.15 

41.32 

54 

82.27 

38.19 

80.92 

39.29 

79.53 

40.35 

78.10 

41.35 

56 

82.23 

38.23 

80.87 

39.33 

79.48 

40.38 

78.06 

41.39 

58 

82.18 

38.26 

80.83 

39.36 

79.44 

40.42 

78.01 

41.42 

60 

82.14 

38.30 

80.78 

39.40 

79.39 

40.45 

77.96 

41 .45 

c-f/  = .75 

.68 

.31 

.68 

.32 

.67 

.33 

.66 

.35 

c+f=  1.00 

.91 

.41 

.90 

.43 

.89 

.45 

.89 

.46 

C+/=  L25 

1.14 

.52 

1.13 

.54 

1.12 

.56 

1.11 

.58 

Table  XI, 


LOGARITHMS  OF  NUMBERS 

FROM 

1 to  10  000 

TO  SIX  DECIMAL  PLACES. 


N. 

Log. 

N. 

Log. 

N. 

Log. 

N. 

Log. 

N. 

Log. 

1 

0.000000 

21 

1.322219 

41 

1.612784 

61 

1.785330 

81 

1.908485 

2 

0.301030 

22 

1.342423 

42 

1.623249 

62 

1.792392 

82 

1.913814 

3 

0.477121 

23 

1.361728 

43 

1.633468 

63 

1.799341 

83 

1.919078' 

4 

0.602060 

24 

1.380211 

44  ' 

1.643453 

64 

1.806180 

84 

1.924279 

5 

0.698970 

25 

1.397940 

45 

1.653213 

65 

1.812913 

85 

1.929419 

6 

0.778151 

26 

1.414973 

46 

1.662758 

66 

1.819544 

86 

1.934498 

7 

0.845098 

27 

1.431364 

47 

1.672098 

67 

1.826075 

87 

1.939519 

8 

0.903090 

28 

1.447158 

48 

1.681241 

68 

1.832509 

88 

1.944483 

9 

0.954243 

' 29 

1.462398 

49 

1.690196 

69 

1.838849 

89 

1.949390 

10 

1.000000 

30 

1.477121  j 

50 

1.698970  ! 

70 

1.845098 

90 

1.954243 

11 

1.041393 

31 

1.491362  ' 

51 

1.707570  i 

71 

1.851258 

91 

1.959041 

12 

1.079181 

32 

1.505150 

52 

1.716003 

72 

1.857332 

92 

1.963788 

13 

1.113943 

33 

1.518514 

53 

1.724276 

73 

1.863323 

93 

1.968483 

14 

1.146128 

34 

1.531479 

54 

| 1 . 732394 

74 

1.869232 

94 

1.973128 

15 

1.176091 

35 

1.544068 

55 

1.740363 

75 

1.875061 

95 

1.977724 

16 

1 204120 

36 

1.556303 

56 

1.748188 

76 

1.880814 

96 

1.982271 

17 

1.230449 

37 

1.568202 

57 

1.755875 

77 

1.886491 

97 

1.986772 

18 

1.255273 

38 

1.579784 

58 

1.763428 

78 

1.892095 

98 

1.991226 

19 

1.278754 

39 

1.591065 

59 

1.770852 

79 

1.897627 

99 

1.995635 

20 

1.301030 

40 

1.602060 

60 

1.778151 

80 

1.903090 

100 

2.000000 

169 


170 


TABLE  XI.  LOGARITHMS  OF  NUMBERS. 


No.  100  L.  000.]  rNo.  109  L.  040. 


N. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

Diff. 

100 

000000 

0434 

0868 

1301 

1734 

2166 

2598 

3029 

3461 

3891 

432 

1 

4321 

4751 

5181 

5609 

6038 

6466 

6894  ! 

7321 

7748 

8174 

428 

2 

8600 

9026 

9451 

9876 

0300 

0724 

1147 

1570 

1993 

2415 

424 

3 

012837 

3259 

3680 

4100 

4521 

4940 

5360 

5779 

6197 

6616 

420 

4 

7033 

7451 

7868 

8284 

8700 

9116 

9532 

9947 

0361 

0775 

416 

5 

021189 

1603 

2016 

2428 

2841 

3252 

3664 

i 4075 

4486 

4896 

412 

6 

5306 

5715 

6125 

6533 

6942 

7350 

7757 

' 8164 

8571 

8978 

408 

7 

9384 

9789 

0195 

0600 

1004 

1408 

1812 

! 2216 

2619 

3021 

404 

8 

033424 

3826 

4227 

4628 

5029 

5430 

5830 

; 6230 

6629 

7028 

400 

9 

7426 

7825 

8223 

8620 

9017 

9414 

9811 

04 

0207 

1 0602 

0998 

397 

Proportional  Parts. 


Diff. 

1 

2 

3 

4 

5 

6 

7 

8 

9 

434 

43.4 

86.8 

130.2 

173.6 

217.0 

260.4 

303.8 

347.2 

390.6 

433 

43.3 

86.6 

129.9 

173.2 

216.5 

259.8 

303.1 

346.4 

389.7 

432 

43.2 

86.4 

129.6 

172.8 

216.0 

259.2 

302.4 

£45.6 

388.8 

431 

43.1 

86.2 

129.3 

172.4 

215.5 

258.6 

301.7 

344.8 

387.9 

430 

43.0 

86.0 

129.0 

172.0 

215.0 

258.0 

301.0 

344.0 

387.0 

429 

42.9 

85.8 

128.7 

in. 6 

214.5 

257.4 

300.3 

343.2  | 

386.1 

428 

42.8 

85.6 

128.4 

171.2 

214.0 

256.8 

299.6 

342.4 

.385.2 

427 

42.7 

85.4 

128.1 

170.8 

213.5 

256.2 

298.9 

341.6 

384.3 

426 

42.6 

85.2 

127.8 

170.4 

213.0 

255.6 

298.2 

340.8 

383.4 

425 

42.5 

85.0 

127.5 

170.0 

212.5 

255.0 

297.5 

340.0 

382.5 

424 

42.4 

84.8 

127.2 

169.6 

212.0 

254.4 

296.8 

339.2 

381.6 

423 

42.3 

84.6 

126.9 

169.2 

211.5 

253.8 

296.1 

338.4 

380.7 

422 

42.2 

84.4 

126.6 

168.8 

211.0 

253.2 

295.4 

337.6 

379.8 

421 

42.1 

84.2 

126.3 

168.4 

210.5 

252.6 

294.7 

336.8 

378.9 

420 

42.0 

84.0 

126.0 

168.0 

210.0 

252.0 

294.0 

336.0 

378.0 

419 

41.9 

83.8 

125.7 

167.6 

209.5 

251.4 

293.3 

335.2 

377.1 

418 

41.8 

83.6 

125.4 

167.2 

209.0 

250.8 

292.6 

334.4 

376.2 

417 

41.7 

83.4 

125.1 

166.8 

208.5 

250.2 

291.9 

333.6 

375.3 

416 

41.6 

83.2 

124.8 

166.4 

208.0 

249.6 

291.2 

332.8 

374.4 

415 

41.5 

83.0 

124.5 

166.0 

207.5 

249.0 

290.5 

332.0 

373.5 

414 

41.4 

82.8 

124.2 

165.6 

207.0 

248.4 

289.8 

331.2 

372.6 

413 

41.3 

82.6 

123.9 

165.2 

206.5 

247.8 

289.1 

330.4 

371.7 

412 

41.2 

82.4 

123.6 

164.8 

206.0 

247.2 

288.4 

329.6 

370.8 

411 

41.1 

82.2 

123.3 

164.4 

205.5 

246.6 

287.7 

328.8 

369.9 

410 

41.0 

82.0 

123.0 

164.0 

205.0 

246.0 

287.0 

328.0 

369.0 

409 

40.9 

81.8 

| 122.7 

163.6 

204.5 

245.4 

286.3 

327.2 

368.1 

408 

40  8 

81.6 

122.4 

163.2 

204.0 

244.8 

285.6 

326.4 

367.2 

407 

40.7 

81.4 

122.1 

162.8 

203.5 

244.2 

284.9 

325.6 

366.3 

406 

40.6 

81.2 

121.8 

162.4 

203.0 

243  6 

284.2 

324.8 

365.4 

405 

| 40.5 

81.0 

121.5 

162.0 

202.5 

243.0 

283.5 

324.0 

| 364.5 

404 

40.4 

80.8 

121.2 

161.6 

202.0 

242.4 

282.8 

323.2 

363.6 

403 

40.3 

80.6 

120.9 

161.2 

201.5 

241.8 

282.1 

322.4 

362.7 

402 

40.2  1 80.4 

120.6 

160.8 

201.0 

241  2 

281.4 

321.6 

i 361.8 

401 

40.1 

80.2 

120.3 

160.4 

200.5 

240.6 

280.7 

320.8 

j 360.9 

400 

40.0 

80-0 

120.0 

160.0 

200.0 

240.0 

280.0 

320.0 

360.0 

399 

39.9 

79.8 

119.7 

159.6 

199.5 

239.4 

279.3 

319.2 

359.1 

398 

39.8 

79.6 

119.4 

159.2 

199.0 

238.8 

278.6 

318.4 

358.2 

397 

39.7 

79.4 

119.1 

158.8 

198.5 

| 238.2 

277.9 

317.6 

357.3 

396 

39.6 

79.2 

118.8 

158.4 

198.0 

237.6 

277.2 

316.8 

356.4 

395 

39.5 

79.0 

118.5 

158.0 

197.5 

I 237.0 

276.5 

316  0 

355.5 

TABLE  XI.  LOGARITHMS  OF  NUMBERS, 


in 


N. 

0 

1 

2 

8 

4 

5 

6 

7 

8 

9 

Diff. 

110 

041393 

1787 

2182 

2576 

2969 

3362 

3755 

4148 

4540 

4932 

393 

1 

5323 
• 9218 

5714 

9606 

6105 

9993 

6495 

6885 

7275 

7664 

8053 

8442 

8830 

390 

0380 

0766 

1153 

1538 

1924 

2309 

2694 

386 

3 

053078 

3463 

3846 

4230 

4613 

4996 

5378 

5760 

6142 

6524 

383 

4 

0905 

7286 

7666 

8046 

8426 

8805 

9185 

9563 

9942 

0320 

379 

5 

060698 

1075 

1452 

1829 

2206 

2582 

2958 

33,33 

3709 

4083 

376 

6 

4458 

4832 

5206 

5580 

5953 

6326 

6699 

7071 

7443 

7815 

373 

7 

8186 

8557 

8928 

9298 

9668 

0038 

0407 

0776 

1145 

1514 

370 

8 

071882 

2250 

2617 

2985 

3352 

3718 

4085 

4451 

4816 

5182 

366 

9 

5547 

5912 

6276 

6640 

7004 

7368 

7731 

8094 

8457 

8819 

363 

Proportional  Parts. 


Diff. 

1 

2 

3 

4 

5 

6 

7 

8 

9 

395 

39.5 

79.0 

118.5 

158.0 

197.5 

237.0 

276.5 

316.0 

355.5 

.394 

39.4 

78.8 

118.2 

157.6 

197.0 

236.4 

275.8 

315.2  ! 

354.6 

393 

39.3 

78.6 

117.9 

157.2 

196.5 

235.8 

275.1 

314.4  i 

353.7 

392 

39.2 

78.4 

117.6 

156.8 

196.0 

235.2 

274.4 

313.6  ; 

352.8 

391 

39.1 

78.2 

117.3 

156.4  | 

195.5 

234.6 

273.7 

312.8 

351.9 

390 

39.0 

78.0 

117.0 

156.0 

195.0 

234.0 

273.0 

312.0 

351.0 

389 

33.9 

77.8 

116.7 

155.6 

194.5 

233.4 

272.3 

311.2  | 

350.1 

388 

38.8 

77.6 

116.4 

155.2 

194.0 

232.8 

271.6 

310.4 

349.2 

387  . 

38.7 

77.4 

116.1 

154.8 

193.5 

232.2 

270.9 

809.6 

348.3 

386 

38.6 

77.2 

115.8 

154.4 

193.0 

231.6 

270.2 

308.8 

347.4 

385 

38.5 

77.0 

115.5 

154.0 

192.5 

231.0 

269.5 

308.0  j 

346.5 

384 

38.4 

76.8 

115.2 

153.6 

192.0 

230.4 

268.8 

307.2  | 

345.6 

383 

38.3 

76.6 

114.9 

153.2 

191.5 

229.8 

268.1 

306.4  ! 

344.7 

382 

38.2 

76.4 

114.6 

152.8 

191.0 

229.2 

267.4 

305.6 

343.8 

381 

38.1 

76.2 

114.3 

152.4 

190.5 

228.6 

266.7 

804.8 

342.9 

380 

38.0 

76.0 

114.0 

152.0 

190.0 

228.0 

266.0 

304.0 

342.0 

379 

37.9 

75.8 

113.7 

151.6 

189.5 

227.4 

265.3 

803.2 

341.1 

378 

37.8 

75.6 

113.4 

151.2 

. 189.0 

226.8 

264.6 

302.4 

340.2 

377 

37.7 

75.4 

113.1 

150.8 

188.5 

'226.2 

263.9 

301.6 

339.3 

376 

37.6 

75.2 

112.8 

150.4 

188.0 

225.6 

263.2 

300.8 

338.4 

375 

37.5 

75.0 

112.5 

150.0 

187.5 

225.0 

262.5 

300.0 

337.5 

374 

37.4 

74  r8 

112.2 

149.6 

187.0 

224.4 

261.8 

299.2 

336.6 

373 

37.3 

74.6 

111.9 

149.2 

186.5 

223.8 

261.1 

288.4 

835.7 

372 

37.2 

74.4 

111.6 

148.8 

186.0 

223.2 

260.4 

297.6 

334.8 

371 

37.1 

74.2 

111.3 

148.4 

185.5 

222.6 

259.7 

286.8 

833.9 

370 

37.0 

74.0 

111.0 

148.0 

185.0 

222.0 

259.0 

286.0 

833 . i ) 

369 

36.9 

73.8 

110.7 

147.6 

184.5 

221.4 

258.3 

295.2 

832.1 

368 

36.8 

73.6 

110.4 

147.2 

184.0 

220.8 

257.6 

294.4 

381.2 

367 

36.7 

73.4 

110.1 

146.8 

183.5 

• 220.2 

256.9 

293.6 

350.3 

- 366 

36.6 

73.2 

109.8 

146.4 

183.0 

219.6 

256.2 

292.8 

329.4 

365 

36.5 

73.0 

109.5 

146.0 

182.5 

219.0 

255.7 

292.0 

828.5 

364 

36.4 

72.8 

109.2 

145.6 

182.0 

218.4 

254.8 

291.2 

327.6 

363 

36.3 

72.6 

108.9 

145.2 

181.5 

217.8 

254.1 

290.4 

326.7 

362 

36.2 

72.4 

108.6 

144.8 

181.0 

217.2 

253.4 

289.6 

325.8 

361 

36.1 

72.2 

108.3 

144.4 

180.5 

216.6 

252.7 

288.8 

324.9 

360 

36.0 

72.0 

108.0 

144.0 

180.0 

216.0 

252.0 

| 288.0 

324.0 

359 

35.9 

71.8 

107.7 

143.6 

179.5 

215.4 

251.3 

287.2 

823.1 

358 

35.8 

71.6 

107.4 

143.2 

179.0 

214.8 

250.6 

286.4 

322.2 

357 

35.7 

71.4 

107.1 

142.8 

178.5 

214.2 

249.9 

285.6 

321.3 

356 

35.6 

71.2 

106.8 

142.4 

178.0 

213.6 

249.2 

284.8 

320.4 

172  TABLE  XI.  LOGARITHMS  OF  NUMBERS. 


No.  120  L.  079.]  [No.  134  L.  130.  I 


N. 

0 

1 

2 

3 

4 

5 

1 6 

7 

8 

9 

Diff. 

120 

079181 

9543 

9904 

i 

0266 

0626 

1 0987 

1347 

1707 

2067 

2426 

360 

1 

082785 

3144 

3503 

3861 

4219  ! 

4576 

4934 

5291 

5647 

60041 

357 

2 

6360 

6716 

7071 

7426 

7781 

8136 

8490 

8845 

9198 

9552 

355 

3 

9905 

0258 

0611 

0963 

1315 

1667 

2018 

2370 

2721 

3071 

352 

4 

093422 

3772 

4122 

4471 

4820 

5169 

5518 

5866 

6215 

6562 

349 

5 

6910 

7257 

7604 

! 7951 

8298 

8644 

8990 

9335 

9681 

0026 

346 

6 

100371 

0715 

1059 

1403 

1747 

2091 

2434 

2777 

3119 

3462 

343 

7 

3804 

4146 

4487 

j 4828 

5169 

5510 

5851 

6191 

6531 

6871 

341 

8 

7’210 

7549 

7888 

! 8227 

8565 

8903 

9241 

9579 

9916 

0253 

338 

9 ; 

110590 

0926 

1263 

1599 

1934 

2270 

2605 

2940 

3275 

3609 

335 

130 

3943 

4277 

4611 

4944 

5278 

! 5611 

5943 

6276 

6608 

6940 

333 

i | 

7271 

•7603 

7934 

8265 

8595 

8926 

9256 

9586 

9915 

0245 

330 

2 i 

120574 

0903 

1231 

1560 

1888 

2216 

2544 

2871 

3198 

3525 

328 

3 

3852 

4178 

4504 

4830 

5156 

! 5481 

5806 

6131 

6456 

6781 

325 

4* 

7105 

7429 

7753 

8076 

8399 

! 8722 

9045 

9368 

9690 

13 

0012 

323 

Proportional  Parts. 


Diff. 

1 

2 

3 

4 

5 

6 

1 r> 

1 

8 

9 

355 

35.5 

71.0 

106.5 

142.0 

177.5 

213.0 

248.5 

284.0 

319.5 

354 

35.4 

70.8 

106.2 

141.6 

177.0 

| 212.4 

! 247.8 

283.2 

318.6 

353 

35.3 

70.6 

105.9 

141.2 

176.5 

i 211.8 

I 247.1 

282.4 

.317.7 

352 

35.2 

70.4 

105.6 

140.8 

176.0 

i 211.2 

246.4 

281.6 

316.8 

351 

35.1 

70.2 

105.3 

140.4 

175.5 

| 210.6 

! 245.7 

280.8 

315.9 

350 

35.0 

70.0 

105.0 

140.0 

175.0 

210.0 

245.0 

280.0 

315.0 

349 

34.9 

69.8 

104.7 

139.6 

174.5 

209.4 

244.3 

279.2 

314.1 

348 

34.8 

69.6 

104.4 

139.2 

174.0 

208.8 

243.6 

278.4 

313.2 

317 

34.7 

69.4 

104.1 

138.8 

173.5 

! 208.2 

242.9 

277 . 6 

312.3 

346 

34.6 

69.2 

103.8 

138.4 

173.0 

207.6 

242.2 

276.8 

311.4 

345 

34.5 

69.0 

103.5 

138.0 

172.5 

207.0 

241.5 

276.0 

310.5 

344 

34.4 

68.8 

103.2 

137.6 

172.0 

206.4 

240.8 

275.2 

309.6 

343a 

34.3 

68.6 

102.9 

137.2 

171.5 

205.8 

240.1 

274.4 

308.7 

342 

34.2 

68.4 

102.6 

136.8 

171.0 

205  2 

239.4 

273.6 

307.8 

341 

34.1 

68.2 

102.3 

136.4 

170.5 

204.6 

238.7 

272.8 

306.9 

310 

34.0 

68.0 

102.0 

136.0 

170.0 

204.0 

238.0 

272.0 

306.0 

339 

33.9 

67.8 

101.7 

135.6 

169.5 

203.4 

237.3 

271.2 

305.1 

833 

33.8 

67.6 

101.4 

135.2 

109.0 

202.8 

236.6 

270.4 

304.2 

337 

33.7 

67.4 

101.1 

134.8 

168.5 

202.2 

235.9 

269.6 

303.3 

336 

33.6 

67.2 

100.8 

134.4 

- 168.0 

201.6 

235.2 

268.8 

302.4 

335 

33.5 

67.0 

100.5 

134.0 

167.5 

201.0 

234.5 

268.0 

301.5 

334 

33.4 

66.8 

100.2 

133.6 

167.0 

200.4 

233.8 

267.2 

300.6 

333 

33.3 

66.6 

99.9 

133.2 

166.5 

199.8 

233.1 

266.4 

299.7 

332 

33.2 

66.4 

99.6 

132.8 

166.0 

199.2 

232.4 

265.6 

298.8 

331 

33.1 

66.2 

99.3 

132.4 

165.5 

198.6 

231.7 

264.8 

297.9 

330 

33.0 

66.0 

99.0 

132.0 

165.0 

198.0 

231.0 

264.0 

297.0 

329 

32.9 

65.8 

98.7 

131.6 

164.5 

197.4 

230.3 

263.2 

296.1 

328 

32.8 

65.6 

98.4 

131.2 

164.0 

196.8 

229.6 

262.4 

295.2 

327 

32.7 

65.4 

98.1 

130.8 

163.5 

196.2 

228.9 

261.6 

294.3 

326 

32.6 

65.2 

97.8 

130.4 

.163.0 

195.6 

228.2 

260.8 

293.4 

325 

32.5 

65.0 

97.5 

130.0 

162.5 

195.0 

227.5 

260.0 

292.5 

324 

32.4 

64.8 

97.2 

129.6 

162.0 

194.4 

226.8 

259.2 

291.6 

323 

32.3 

64.6 

96.9 

129.2 

161.5 

193.8 

226.1 

258.4 

290.7 

322 

32.2 

64.4 

96.6 

128.8 

161.0 

193.2 

225.4 

257.6  | 

289.8 

TABLE  XT.  LOGARITHMS  OF  NUMBERS, 


173 


No.  135  L.  130.]  [No.  149  L.  175. 


N. 

0 

1 

2 

3 

4 

5 I 

6 

7 

8 

9 

Diff. 

135 

130334 

0655 

0977 

1298 

1619 

1939 

2260 

2580  ! 

2900 

3219 

321 

G 

3539 

3858 

4177 

4496 

4814 

5133  1 

5451 

5769 

6086 

6403 

318 

7 

8 

6721 

9879 

7037 

7354 

7671 

7987 

8303 

8618 

8934  | 

9249 

9564 

316 

0194 

0508 

0822 

1136 

1450 

| 1763 

2076 

2389 

2702 

314 

9 

143015 

3327 

3639 

3951 

4263 

4574 

j 4885 

5196 

5507 

5818 

311 

140 

1 

6128 

9219 

6438 

9527 

6748 

9835 

7058 

7367 

7676 

7985 

8294 

8603 

8911 

309 

0142 

0449 

0756 

1063 

1370 

1676 

1982 

307 

2 

152288 

2594 

2900 

3205 

3510 

3815 

4120 

4424 

47'28 

5032 

305 

3 

5336 

5640 

5943 

6246 

6549 

6852 

7154 

7457 

7759 

8061 

303 

A 

8362 

8664 

8965 

9266 

9567 

9868 

‘t 

0168 

3161 

0469 

3460 

0769 

3758 

1068 

4055 

301 

299 

5 

161368 

1667 

1967 

2266 

2564 

2863 

6 

4353 

4650 

4947 

5244 

5541 

5838 

6134 

6430 

6726 

7022 

297 

J 

7317 

7613 

7908 

8203 

8497 

8792 

9086 

9380 

9674 

9968 

295 

8 

170262 

0555 

0848 

1141 

1434 

1726 

i 2019 

2311 

2603 

2895 

293 

9 

3186 

3478 

3769 

4060 

4351 

4641 

4932 

5222 

5512 

5802 

291 

Proportional  Parts. 


Diff. 

i 

2 

3 

4 

5 

6 

7 

8 

9 

321 

32.1 

64.2 

96.3 

128.4 

160.5 

192.6  ; 

224.7 

256.8 

288.9 

320 

32.0 

64.0 

96.0 

128.0 

160.0 

192.0 

224.0 

256.0 

288.0 

319 

31.9 

63.8 

95.7 

127.6 

159.5 

191.4 

223.3 

255.2 

287.1 

318 

31.8 

63.6 

95.4 

127.2 

159  0 

190.8 

222.6 

254.4 

286.2 

317 

31.7 

63.4 

95.  i 

128.8 

158.5 

190.2 

221.9 

253.6 

285.3 

316 

31.6 

63.2 

94.8 

126.4 

158.0 

189.6 

221.2 

252.8 

284.4 

315 

31 .5 

63.0 

94.5 

126.0 

157.5 

189.0 

220.5  1 

252.0 

283.5 

314 

31.4 

62.8 

94.2 

125.6 

157.0 

188.4 

219.8  I 

251.2 

282.6 

313 

31.3 

62.6 

93.9 

125.2 

156.5 

187.8 

219.1  1 

250.4 

281.7 

312 

31.2 

62.4 

93.6 

124.8 

156.0 

187.2 

218.4 

249.6 

280.8  ! 

311 

31.1 

62.2 

93.3 

124.4 

155.5 

186.6 

217.7  i 

248.8 

279.0  ! 

: 310 

31.0 

62.0 

93.0 

124.0 

155.0 

186.0 

217.0 

248.0 

279.0  i 

| 309 

30.9 

61.8 

92.7 

123.6 

154.5 

185.4 

216.3 

247.2 

278.1 

! 308 

30.8 

61.6 

92.4 

123.2 

154.0 

184.8 

215.6 

246.4 

277.2 

307 

30.7  j 

61.4 

92.1 

122.8 

153.5 

184.2 

214.9 

245.6 

276.3 

306 

30.6  j 

61.2 

91.8 

122.4 

153.0 

183.6 

214.2 

244.8 

275.4 

305 

30.5 

61.0 

91.5 

122.0 

152.5 

183.0 

213.5 

244.0 

274,5 

304 

30.4 

60.8 

91.2 

121.6 

152.0 

182.4 

212.8 

243.2 

273.6 

303 

30.3  i 

60.6 

90.9 

121.2 

151.5 

181.8 

212.1 

242.4 

272.7 

302 

30.2 

60.4 

90.6 

120.8 

151.0 

181.2 

211.4 

241.6 

.271.8 

301 

30.1 

1 60.2 

90.3 

120.4 

150.5 

180.6 

210.7 

240.8 

270.9 

300 

30.0 

60.0 

90.0 

120.0 

150.0 

180.0 

210.0 

240.0 

270.0 

299 

29.9 

i 59.8 

89.7 

119.6 

149.5 

179.4 

209.3 

239.2 

269.1 

298 

29.8 

| 59.6 

89.4 

119.2 

149.0 

178.8 

208.6 

.238.4 

268.2 

297 

29.7 

i 59.4 

89.1 

118.8 

148.5 

178.2 

'207.9 

237.6 

267.3 

296 

29.6 

59.2 

88.8 

118.4 

148.0 

177.6 

207.2 

236.8 

266.4 

295 

29.5 

59.0 

88.5 

118.0 

147.5 

177.0 

206.5 

236.0 

265.5 

294 

29.4 

58.8 

88.2 

117.6 

147.0 

176.4 

205.8 

235.2 

264.6 

293 

29.3 

58.6 

87.9 

117.2 

146.5 

175.8 

205.1 

234.4 

263.7 

292 

29.2 

58.4 

87.6 

116.8 

146.0 

175.2 

2^4.4 

233.6 

262.8 

291 

29.1 

58.2 

87.3 

116.4 

145.5 

174.6 

203.7 

232.8 

261.9 

290 

29.0 

58.0 

87.0 

116.0 

145.0 

174.0 

203.0 

232.0 

261.0 

289 

28.9 

57.8 

86.7 

115.6 

144.5 

173.4 

202.3 

231.2 

260.1 

288 

28.8 

57.6 

86.4 

115.2 

144.0 

172.8 

201.6 

230.4 

259.2 

287 

28.7 

57.4 

86.1 

114.8* 

143.5 

172.2 

200.9 

229.6 

258.3 

286 

28.6 

57.2 

85.8 

114.4 

143.0 

171.6 

200.2 

228.8 

257.4 

174 


TABLE  XL  LOGARITHMS  OF  NUMBERS, 


No.  150  L.  176.1  [No.  169  L.  230. 


N. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

Diff. 

150 

176091 

6381 

6o70 

6959 

7248 

7536 

7825 

8113 

8401 

8689 

289 

l 

8977 

9264 

9552 

9839 

0126 

0413 

0699 

0986 

1272 

1558 

287 

2 

181844 

2129 

2415 

2700 

2985 

3270 

3555 

3839 

4123 

4407 

285 

3 

4691 

4975 

5259 

5542 

5825 

6108 

6591 

6674 

6956 

7239 

283 

4 

7521 

7803 

8084 

8366 

8647 

8928 

9209 

9490 

9771 

0051 

281 

5 

190332 

0612 

0892 

1171 

1451 

1730 

2010 

2289 

2567 

2846 

279 

6 

3125 

3403 

3681 

3959 

4237 

4514 

4792 

5069 

5346 

5623 

278 

7 

8 

5900 

8657 

6176 

8932 

6453 

9206 

6729 

9481 

7005 

9755 

7281 

7556 

7832 

8107 

8382 

276 

0029 

0303 

0577 

0850 

1124 

274 

9 

201397 

1670 

1943 

2216 

| 2488 

2761 

3033 

3305 

3577 

3848 

272 

160 

4120 

4391 

4663 

4934 

5204 

5475 

5746 

6016 

6286 

6556 

271 

1 

2 

6826 

9515 

7096 

9783 

7365 

7634 

| 7904 

8173 

8441 

8710 

8979 

9247 

269 

0051 

0319 

0586 

0853 

1121 

1388 

1654 

1921 

267 

3 

212188 

2454 

2720 

2986 

3252 

3518 

3783 

4049 

4314 

4579 

266 

4 

4844 

5109 

5373 

5638 

5902 

6166 

6430 

6694 

6957 

7221 

264 

5 

7484 

7747 

8010 

8273 

8536 

8798 

9060 

9323 

9585 

9846 

262 

6 

220108 

0370 

: 0631 

0892 

1153 

1414 

1675 

1936 

2196 

2456 

261 

7 

2716 

i 2976 

3236 

3496 

i 3755 

4015 

4274 

4533 

4792 

5051 

259 

8 

9 

5309 

7887 

5568 
| 8144 

1 5826 
| 8400 

6084 

8657 

i 6342 
! 8913 

! 6600 
1 9170 

6858 

9426 

• 7115 
9682 

7372 

9938 

7630 

258 

23 

j 

1 

1 

0193 

256 

Proportional  Parts. 


Diff. 

1 

2 

3 

4 

5 

6 

7 

8 

9 

285 

28.5 

57.0 

85.5 

114.0 

142.5 

171.0 

199.5 

228.0 

256.5 

284 

28.4 

56.8 

85.2 

113.6 

142.0 

170.4 

198.8 

227.2 

255.6 

203 

28.3 

56.6 

84.9 

113.2 

141.5 

169.8 

198.1 

226.4 

254.7 

282 

28.2 

56.4 

84.6 

112.8 

141.0 

169.2 

197.4 

225.6 

253.8 

201 

28.1 

56.2 

84.3 

112  4 

140.5 

168.6 

196.7 

224.8 

252.9 

200 

28.0 

56.0 

84.0 

112.0 

140.0 

168.0 

196.0 

224.0 

252.0 

279 

27.9 

55.8 

83.7 

111.6 

139.5 

167.4 

195.3 

223.2 

251.1 

278 

27.8 

55.6 

83.4 

111.2 

139.0 

166.8 

194.6 

222.4 

250.2 

27'7 

27.7 

55.4 

83.1 

110.8 

138.5 

1 166.2 

193.9 

221.6 

249.3 

276' 

27.6 

55.2 

82.8 

110.4 

138.0- 

! 165.6 

193.2 

220.8 

248.4 

275 

27.5 

55.0 

82.5 

110.0 

137.5 

165.0 

192.5 

220.0 

247.5 

274 

27.4 

54.8 

82.2 

109.6 

137.0 

j 164.4 

191.8 

219.2 

246.6 

273 

27.3 

54.6 

81.9 

109.2 

136.5 

163.8 

191.1 

218.4 

245.7 

272 

27.2 

54.4 

81.6 

108.8 

136.0 

163.2 

190.4 

217.6 

244.8 

271 

27.1 

54.2 

81.3 

108.4 

135.5 

162.6 

189.7 

216.8 

243.9 

270 

27.0 

54.0 

81.0 

108.0 

135.0 

162.0 

189.0 

216.0 

243.0 

269 

26.9 

53.8 

80.7 

107.6 

134.5 

161:4 

188.3 

215.2 

242.1 

268 

26.8 

53.6 

80.4 

107.2 

134.0 

i 160.8 

187.6 

214.4 

241.2 

267 

26.7 

53.4 

80.1 

106.8 

133.5 

i 160.2 

186.9 

213.6 

240.3 

266 

26.6 

53.2 

79.8 

106.4 

133.0 

159.6 

186.2 

212.8 

239.4 

265 

26.5 

53.0 

79.5 

106.0 

132.5 

159.0 

185.5  ! 

212.0 

238.5 

264 

26.4 

52.8 

79.2 

105.6 

132.0 

158.4 

184.8 

211.2 

237.6 

263 

26.3 

52.6 

78.9 

105.2 

131.5 

157.8 

184.1 

210.4 

236.7 

262 

26.2 

52.4 

78.6 

104.8 

131.0 

157.2 

183.4 

209.6 

235.8 

261 

26.1 

52.2 

78.3 

104.4 

130.5 

156.6 

182.7 

208.8 

234.9 

260 

26.0 

52.0 

78.0 

104.0 

130.0 

156.0 

182.0 

208.0 

234.0 

259 

25.9 

51.8 

77.7 

103.6 

129.5 

155.4 

181.3 

207.2 

233.1 

258 

25.8 

51.6 

77.4 

103.2 

'129.0 

154.8 

180.6 

206.4 

232.2 

257 

25.7 

51.4 

77.1 

102.8 

128.5 

154.2  j 

179.9 

205.6 

231.3 

256 

25.6 

51.2 

76.8 

102.4 

128.0 

153.6 

179.2 

204.8 

230.4 

255 

25.5 

51.0 

76.5 

102.0 

127.5 

153.0  | 

178.5 

204.0 

229.5 

TABLE  XI.  LOGARITHMS  OF  NUMBERS. 


175 


No.  170  L.  230.]  [No.  189  L.  278. 


N. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

Diff. 

170 

230449 

0704 

0960 

1215 

1470 

1724 

1979 

2234 

2488 

2742 

255 

1 

2996 

3250 

3504 

3757 

4011 

4264 

4517 

4770 

5023 

5276 

253 

2 

5528 

5781 

6033 

6285 

6537 

6789 

7041 

7292 

7544 

7795 

252 

3 

8046 

8297 

8548 

8799 

9049 

9299 

9550 

9800 

0050 

0300 

250 

4 

240549 

0799 

1048 

1297 

1546 

1795 

2044 

2293 

2541 

2790 

249 

5 

3038 

3286 

3534 

3782 

4030 

4277 

4525 

4772 

5019 

5266 

248 

6 

5513 

5759 

6006 

6252 

6499 

6745 

6991 

7237 

7482 

7728 

246 

7 

7973 

8219 

8464 

8709 

8954 

9198 

9443 

9687 

9932 

0176 

245 

8 

250420 

0664 

0908 

1151 

1395 

1638 

1001 

2125 

2368 

2610 

243 

9 

2853 

3096 

3338 

3580 

3822 

4064 

4306 

4548 

4790 

5031 

242 

180 

5273 

5514 

5755 

5996 

6237 

6477 

6718 

6958 

7198 

7439 

241 

1 

7679 

7918 

8158 

8398 

8637 

8877 

9116 

9355 

9594 

9833 

239 

2 

260071 

0310 

0548 

0787 

1025 

1263 

1501 

1739 

1976 

2214 

238 

3 

2451 

2688 

2925 

3162 

3399 

3636 

3873 

4109 

<1346 

4582 

237 

4 

4818 

5054 

5290 

5525 

5761 

5996 

6232 

6467 

6702 

6937 

235 

5 

7172 

7406 

7641 

7875 

8110 

8344 

8578 

8812 

9046 

9279 

234 

g 

9513 

9746 

9980 

0213 

0446 

0679 

0912 

1144 

1377 

1609 

233 

7 

271842 

2074 

2306 

2538 

2770 

3001 

3233 

3464 

3696 

3927 

232 

8 

4158 

4389 

4620 

4850 

5081 

5311 

5542 

5772 

6002 

6232 

230 

9 

6462 

6692 

6921 

7151 

7380 

7609 

7838 

8067 

8296 

8525 

229 

Proportional  Parts. 


Diff. 

1 

2 

3 

4 

5 

6 

7 

8 

9 

255 

25.5 

51.0 

76.5 

102.0 

127.5 

153.0 

178.5 

204.0 

229.5 

254 

25.4 

50.8 

76.2 

101.6 

127.0 

152.4 

177.8 

203.2 

228.6 

253 

25.3 

50.6 

75.9 

101.2 

126.5 

151.8 

177.1 

202.4 

227.7 

252 

25.2 

50.4 

75.6 

100.8 

126.0 

151.2 

176.4 

201.6 

226.8 

251 

25.1 

50.2 

75.3 

100.4 

125.5 

150.6 

175.7 

200.8 

225.9 

250 

25  0 

50.0 

75.0 

100.0 

125.0 

150.0 

175.0 

200.0 

225.0 

249 

24.9 

49.8 

74.7 

99.6 

124.5 

149.4 

174.3 

199.2 

224.1 

248 

24.8 

49.6 

74.4 

99.2 

124.0 

148.8 

173.6 

198.4 

223.2 

247 

24.7 

49.4 

74.1 

98.8 

123.5 

148.2 

172.9 

197.6 

222.3 

246 

24.6 

49.2 

73.8 

98.4 

123.0 

147.6 

172.2 

196.8 

221.4 

245 

24.5 

49.0 

73.5 

98.0 

122.5 

147.0 

171.5 

196.0 

220.5 

244 

24.4 

48'.  8 

73.2 

97.6 

122.0 

146.4 

170.8 

195.2 

219.6 

243 

24.3 

48.6 

72.9 

97.2 

121.5 

145.8 

170.1 

194.4 

218.7 

242 

24.2 

48.4 

72.6 

96.8 

121.0 

145.2 

169.4 

193.6 

217.8 

241 

24.1 

48.2 

72.3 

96.4 

120.5 

144.6 

168.7 

192.8 

216.9 

240 

24.0 

48.0 

72.0 

96.0 

120.0 

144.0 

168.0 

192.0 

216.0 

239 

23.9 

47.8 

71.7 

95.6 

119.5 

143.4 

167.3 

191.2 

215.1 

238 

23.8 

47.6 

71.4 

95.2 

119  :o 

142.8 

166.6 

190.4 

214.2 

237 

23.7 

47.4 

71.1 

94.8 

118.5 

142.2 

165.9 

189.6 

213.3 

236 

23.6 

47.2 

70.8 

94.4 

118.0 

141.6 

165.2 

188.8 

212.4 

235 

23.5 

47.0 

70.5 

94.0 

117.5 

141.0 

164.5 

188.0 

211.5 

234 

23.4 

46.8 

70.2 

93.6 

117.0 

140.4 

163.8 

187.2 

210.6 

233 

23.3 

46.6 

69.9 

93.2 

116.5 

139.8 

163.1 

186.4 

209.7 

232 

23.2 

46.4 

69.6 

92.8 

116.0 

139.2 

162.4 

185.6 

208.8 

231 

23.1 

46.2 

69.3 

92.4 

115.5 

138.6 

161.7 

184.8 

207.9 

230 

23.0 

46.0 

69.0 

92.0 

115.0 

138.0 

161.0 

184.0 

207.0 

229 

22.9 

45.8 

68.7 

91.6 

114.5 

137.4 

160.3 

183.2 

206.1 

228 

22.8 

45.6 

68.4 

91.2 

114.0 

136.8 

159.6 

182.4 

205.2 

227 

22.7 

45.4 

68.1 

90.8 

113.5 

136.2 

158.9 

181.6 

204.3 

226 

22.6 

45.2 

67.8 

90.4 

113.0 

135.6 

158  2 

180.8 

203.4 

176 


TABLE  XI.  LOGARITHMS  OF  NUMBERS, 


No.  190  L.  278.]  [No.  214  L.  332. 


N. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

Diff. 

190 

278754 

8982 

9211 

9439 

9667 

9895 

0123 

0351 

0578 

0806 

228 

1 

281033 

1261 

1488 

1715 

1942 

2169 

2396 

2622 

2849 

3075 

227 

2 

3301 

3527 

3753 

3979 

4205 

4431 

4656 

4882 

5107 

5332 

226 

3 

v 5557 

5782 

6007 

6232 

6456 

6681 

6905 

7130 

7354 

7578 

225 

4 

7802 

8026 

8249 

8473 

8696 

8920 

9143 

9366 

9589 

9812 

223 

5 

290035 

0257 

0480 

0702 

0925 

1147 

1369 

1591 

1813 

2034 

222 

6 

2256 

2478 

2699 

2920 

3141 

3363 

3584 

3804 

4025 

4246 

221 

7 

4466 

4687 

4907 

5127 

5347 

5567 

5787 

6007 

6226 

6446 

220 

8 

6665 

6884 

7104 

7323 

7542 

7761 

7979 

8198 

8416 

8635 

219 

9 

8853 

9071 

9289 

9507 

9725 

9943 

0161 

0378 

0595 

0813 

218 

200 

301030 

1247 

1464 

1681 

1898 

2114 

2331 

2547 

2764 

2980 

217 

1 

3196 

3412 

3628 

3844 

4059 

4275 

4491 

4706 

4921 

5136 

216 

2 

5351 

5566 

5781 

5996 

6211 

6425 

6639 

6854 

7068 

7282 

215 

3 

7496 

7710 

7924 

8137 

8351 

8564 

8778 

8991 

9204 

9417 

213 

4 

9630 

9843 

0056 

0268 

0481 

0693 

0906 

1118 

1330 

1542 

212 

5 

311754 

1966 

2177 

2389 

2600 

2812 

3023 

3234 

3445 

3656 

211 

6 

3867 

4078 

4289 

4499 

4710 

4920 

5130 

5340 

5551 

5760 

210 

7 

5970 

6180 

6390 

6599 

6809 

, 7018 

7227 

7436 

7646 

7854 

209 

8 

8063 

8272 

8481 

8689 

8898 

9106 

9314 

9522 

9730 

9938 

208 

9 

320146 

0354 

0562 

0769 

0977 

1184 

1391 

1598 

1805 

2012 

207 

210 

2219 

2426 

2633 

2839 

3046 

3252 

3458 

3665 

3871 

4077 

206 

1 

4282 

4488 

4694 

4899 

5105 

5310 

5516 

5721 

5926 

6131 

205 

2 

6336 

6541 

6745 

6950 

7155 

7359 

7563 

7767 

79?'2 

8176 

204 

3 

8380 

8583 

8787 

8991 

9194 

9398 

9601 

9805 

0008 

0211 

203 

4 

330414 

0617 

0819 

1022 

1225 

i 1427 

1630 

1832 

2034 

2236 

202 

Proportional  Parts. 


Diff. 

1 

2 

3 

4 

5 

6 

7 

8 

9 

225 

22.5 

45.0 

67.5 

90.0 

112.5 

135.0 

157.5 

180.0 

202.5 

224 

22.4 

44.8 

67.2 

89.6 

112.0 

134.4 

156.8 

179.2 

201.6 

223 

22.3 

44.6 

66.9 

89.2 

111.5 

133.8 

156.1 

178.4 

200.7 

222 

-22.2 

44.4 

66.6 

88.8 

111.0 

133.2 

155.4 

177.6 

199.8 

221 

22.1 

44.2 

66.3 

88.4 

110.5 

132.6 

154.7 

176.8 

198.9 

220 

22.0 

44.0 

66.0 

88.0 

110.0 

132.0' 

154.0 

176.0 

198.0 

219 

21.9 

43.8 

65.7 

87.6 

109.5 

131.4 

153.3 

175.2 

197.1 

218 

21.8 

43.6 

65.4 

87.2 

109.0 

130.8 

152.6 

174.4 

196.2 

217 

21.7 

43.4 

65.1 

86.8 

108.5 

130.2 

151.9 

173.6 

195.3 

216 

21.6 

43.2 

64.8 

86.4 

108.0 

129.6 

151.2 

172.8 

194.4 

215 

21.5 

43.0 

64.5 

86.0 

107.5 

129.0 

150.5 

172.0 

193.5 

214 

21.4 

42.8 

64.2 

85.6 

107.0 

128.4 

149.8 

171.2 

192.6 

213 

21.3 

42.6 

63.9 

85.2 

106.5 

127.8 

149.1 

170.4 

191.7 

212 

21.2 

42.4 

63.6 

84.8 

106.0 

127.2 

148.4 

169.6 

190.8 

211 

21.1 

42.2 

63.3 

84.4 

105.5 

126.6 

147.7 

168.8 

189.9 

210 

21.0 

42.0 

63.0 

84.0 

105.0 

126.0 

147.0 

168.0 

189.0 

209 

20.9 

41.8 

62.7 

83.6 

104.5 

125.4 

146.3 

167.2 

188.1 

208 

20.8 

41.6 

62.4 

83.2 

104.0 

124.8 

145.6 

166  4 

187.2 

207 

20.7 

41.4 

62.1 

82.8 

103.5 

124.2 

144.9 

165.6 

186.3 

206 

20.6 

41.2 

61.8 

82.4 

103.0 

123.6 

144.2 

164.8 

185.4 

205 

20.5 

41.0 

61.5 

82.0 

102.5 

123.0 

143.5 

164.0 

184.5 

204 

20.4 

40.8 

61.2 

81.6 

102.0 

122.4 

142.8 

163.2 

183.6 

203 

20.3 

40.6 

60.9 

81.2 

101.5 

121.8 

142.1 

102.4 

182.7 

202 

20.2 

40.4 

60.6 

')0.8 

101.0 

121.2 

141.4 

161.6 

181.8 

TABLE  XI.  LOGARITHMS  OF  NUMBERS, 


177 


No.  215  L.  332.]  [No.  239  L.  380. 


N. 

0 

1 

2 

3 

4 

5 

6 

7 ; 

8 

9' 

Diff. 

215 

332438 

2640 

2842 

3044 

3246 

3417 

3649 

3850 

4051 

4253 

202 

G 

4454 

4655 

4856 

50S  7 

5257 

5458 

5658 

5859 

6059 

6260 

201 

7 

G4G0 

6660 

6860 

7060 

7260 

7459 

7659 

7858 

8058 

8257 

200 

8 

8456 

8656 

8855 

9054 

9253 

9451 

9650 

9849 

0047 

0246 

199 

9 

340444 

0642 

0841 

1039 

1237 

1435 

1632 

1830 

2028 

2225 

198 

220 

2423 

2620 

2817 

3014 

3212 

3409 

3606 

3802 

3999 

4196 

197 

1 

4392 

4589 

4785 

4981 

5178 

5374 

5570 

5766 

5962 

i 6157 

196 

2 

G353 

6549 

6744 

6939 

7135 

7330 

7525 

7720 

7915 

8110 

195 

3 

8305 

8500 

8694 

8889 

9083 

9278 

9472 

9666 

9860 

0054 

i (it 

4 

350248 

0442 

0636 

0829 

1023 

1216 

1410 

1603 

1796 

1989 

193 

5 

2183 

2375 

2568 

2761 

2954 

3147 

3339 

3532 

3724 

3916 

193 

G 

4108 

4301 

4493 

4685 

4876 

5068 

5260 

5452 

5643 

5834 

192 

7 

6026 

6217 

6408 

6599 

6790 

6981 

j 7172 

7363 

7554 

7744 

191 

8 

7935 

8125 

8316 

8506 

8696 

8886 

9076 

9266 

9456 

9646 

190 

9 

9835 

0025 

0215 

0404 

0593 

0783 

0972 

1161 

1350 

1539 

189 

230 

361728 

1917 

2105 

2294 

2482 

2671 

2859 

3048 

3236 

3424 

188 

1 

3612 

3800 

3988 

4176 

4363 

1 4551 

4739 

4926 

5113 

5301 

188 

2 

5488 

5675 

5862 

6049 

6236 

6423 

6610 

6796 

6983 

7169 

187 

3 

7356 

7542 

7729 

7915 

8101 

8287 

8473 

8659 

8845 

9030 

186 

4 

9216 

9401 

9587 

9772 

9958 

0143 

0328 

0513 

0698 

0883 

185 

5 

371068 

1253 

1437 

1622 

1806 

1991 

2175 

2360 

2544 

2728 

184 

6 

2912 

3096 

3280 

3464 

3647 

3831 

4015 

4198 

4382 

4565 

184 

7 

4748 

4932 

5115 

5298 

5481 

5664 

5846 

6029 

6212 

6394 

183 

8 

6577 

6759 

6942 

7124 

7306 

7488 

7670 

7852 

8034 

8216 

182 

9 

8398 

8580 

8761 

8943 

9124 

9306 

9487 

9668 

9849 

38 

0030 

181 

Proportional  Parts. 


Diff. 

1 

2 

3 

4 

5 

6 

7 

8 

9 

202 

20.2 

40.4 

60.6 

80.8 

101.0 

121.2 

141.4 

"Tgi.g 

181.8 

201 

I 20.1 

40.2 

60.3 

80.4 

100.5 

120.6 

140.7 

160.8 

1(0.9 

200  ! 

20.0 

40.0 

60.0 

80.0 

100.0 

120.0 

140.0 

160.0 

180.0 

199  | 

19.9 

39.8 

59.7 

79.6 

99.5 

119.4 

139.3 

159.2 

179.1 

198  1 

19.8 

;39.6 

59.4 

79.2 

99.0 

118.8 

138.6 

158.4 

178.2 

197 

19.7 

39.4 

59.1 

78.8 

98.5 

118.2 

137.9 

157.6 

177.3 

196  ! 

19.6 

39.2 

58.8 

78.4 

98.0 

117.6 

137.2 

156.8 

176  4 

195 

19.5 

39.0 

58.5 

78.0 

97  5 

117.0 

136.5 

156.0 

175.5 

194  ; 

19.4 

38.8 

58.2 

77.6 

97.0 

116.4 

135.8 

155.2 

174.6 

193 

19.3 

38.6 

57.9 

77.2 

96.5 

115.8 

135.1 

154.4 

173  7 

192 

1 19.2 

38.4 

57.6 

76.8 

96.0 

115.2 

134.4 

153.6 

172.8 

191 

1 19.1 

38.2 

57.3 

76.4 

95.5 

114.6 

133.7 

152.8 

171.9 

190 

: 19.0 

38.0 

57.0 

76.0 

95.0 

114.0 

133.0 

152.0 

171.0 

189 

j 18.9 

37.8 

56.7 

75.6 

94.5 

113.4 

132.3 

151.2 

170.1 

188 

18.8 

37.6 

56.4 

75.2 

94.0 

112.8 

131.6 

150.4 

169.2 

187 

18.7 

37  4 

56.1 

74.8 

93.5 

112.2 

130.9 

149.6 

168.3 

186 

18.6 

37.2 

55.8 

74.4 

93.0 

111.6 

130.2 

148.8 

167.4 

185 

18.5 

37.0 

55.5 

74.0 

92.5 

111.0 

129.5 

148.0 

166.5 

184 

18.4 

36.8 

55.2 

73.6 

92.0 

110.4 

128.8 

147.2 

165.6 

183 

18.3 

36.6 

54.9 

73.2 

91.5 

109.8 

128.1 

146.4 

164.7 

182 

18.2 

36.4 

54.6 

72.8 

91.0 

109.2 

127.4 

145.6 

163.8 

181 

18.1 

36.2 

54.3 

72.4 

90.5 

108.6 

126.7 

144.8 

162.9 

180 

18.0 

36.0 

54.0 

72.0 

90.0 

108.0 

126.0 

144.0 

162.0 

179 

17.9 

35.8 

53.7 

71.6 

89.5 

107.4 

125.3 

143.2 

161.1 

178 


TABLE  XI.  LOGARITHMS  OF  NUMBERS. 


No.  240  L.  380.]  [No.  209  L.  431. 


N. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

Dili. 

240 

380211 

0392 

0573 

0754 

0934 

1115 

1296 

1476 

1656 

1837 

181 

1 

2017 

2197 

2377 

2557 

2737 

2917 

3097 

3277 

3456 

3636 

180 

2 

3815 

3995 

4174 

4353 

45133 

4712 

4891 

5070 

5249 

5428 

179 

3 

5606 

5785 

5964 

6142 

6321 

6499 

6677 

6856 

7034 

7212 

178 

4 

7390 

7568 

7746 

7924 

8101 

8279 

8456 

8634 

8811 

8989 

178 

9166 

9343 

9520 

9698 

9875 

0051 

1817 

0228 

1993 

0405 

2169 

0582 

2345 

0759 

2521 

177 

176 

6 

390935 

1112 

1288 

1464 

1641 

7 

2697 

2873 

3048 

3224 

3400 

3575 

3751 

3926 

4101 

4277 

176 

8 

4452 

4627 

4802 

4977 

5152 

5326 

5501 

5676 

5850 

6025 

175 

9 

6199 

6374 

6548 

6722 

6896 

7071 

7245 

7419 

7592 

7766 

174 

250 

1 

7940 

9674 

8114 

9847 

8287 

8461 

8634 

8808 

8981 

9154 

9328 

9501 

173 

0020 

0192 

0365 

0538 

0711 

0883 

1056 

1228 

173 

2 

401401 

1573 

1745 

1917 

2089 

2261 

2433 

2605 

2777 

2949 

172 

3 

3121 

3292 

3464 

3635 

.3807 

3978 

4149 

4320 

4492 

4663 

171 

4 

4834 

5005 

5176 

5346 

5517 

5688 

5858 

6029 

6199 

6370 

171 

5 

6540 

6710 

6881 

7051 

7221 

7391 

7561 

7731 

7901 

8070 

170 

6 

8.240 

8410 

8579 

8749 

8918 

9087 

9257 

9426 

9595 

9764 

169 

7 

9933 

0102 

0271 

0440 

0609 

0777 

0946 

1114 

1283 

1451 

169 

8 

411620 

1788 

1956 

2124 

2293 

2461 

2629 

2796 

2964 

3132 

168 

9 

3300 

3467 

3635 

3803 

3970 

4137 

4305 

4472 

4639 

4806 

167 

260 

4973 

5140 

5307 

5474 

5641 

5808 

5974 

6141 

6308 

6474 

167 

1 

6641 

6807 

6973 

7139 

7306 

7472 

7638 

7804 

7970 

8135 

166 

2 

3 

8301 

9956 

8467 

8633 

8798 

8964 

9129 

9295 

9460 

9625 

9791 

165 

0121 

0286 

0451 

0616 

0781 

0945 

1110 

1275 

1439 

165 

4 

421604 

1768 

1933 

2097 

2261 

2426 

2590 

2754 

2918 

3082 

164 

5 

3246 

3410 

3574 

3737 

3901 

4065 

4228 

4392 

4555 

4718 

164 

6 

4882 

5045 

5208 

5371 

5534 

5697 

5860 

6023 

6186 

6319 

163 

7 

6511 

6674 

6836 

6999 

7161 

7324 

7486 

7648 

7811 

7973 

162 

8 

9 

8135 

9752 

8297 

9914 

8459 

8621 

8783 

8944 

9106 

9268 

9429 

9591 

162 

43 

0075 

0236 

0398 

0559 

0720 

0881 

i 1042 

1203 

161 

Proportional  Parts. 


Diff. 

1 

2 

3 

4 

5 

6 

7 

8 

9 

178 

17.8 

35.6 

53.4 

71.2 

89.0 

103.8 

124.6 

142.4 

160.2 

177 

17.7 

35.4 

53.1 

70.8 

88.5 

108.2 

123.9 

141.6 

159.3 

176 

17.6 

35.2 

52.8 

70.4 

88.0 

105.6 

123.2 

140.8 

158.4 

175 

17.5 

35.0 

52.5 

70.0 

87.5 

105.0 

122.5 

140.0 

157.5 

174 

17.4 

34.8 

52.2 

69.6 

87.0 

104.4 

121.8 

139.2 

156.6 

173 

17.3 

34.6 

51.9 

69.2 

86.5 

103.8 

121.1 

138.4 

155.7 

172 

17.2 

34.4 

51.6 

68.8 

86.0 

103.2 

120.4 

137.6 

154.8 

171 

17.1 

34.2 

51.3 

68.4 

85.5 

102.6 

119.7 

136.8 

153.9 

170 

17.0 

34.0 

51.0 

68.0 

85.0 

102.0 

119.0 

136.0 

153.0 

139 

16.9 

33.8 

50.7 

67.6 

84.5 

101.4 

118.3 

135.2 

152.1 

138 

16.8 

33.6 

50.4 

67.2 

84.0 

100.8 

117.6 

134  4 

151.2 

137 

16.7 

33.4 

50.1 

66.8 

83.5 

100.2 

116.9 

133.6 

150.3 

136 

16.6 

33.2 

49.8 

66.4 

83.0 

99.6 

116.2 

132.8 

149.4 

1 35 

16.5 

33.0 

49.5 

66.0 

82.5 

99.0 

115.5 

132.0 

148.5 

164 

16.4 

32.8 

49.2 

65.6 

82.0 

98.4 

114.8 

131.2 

147.6 

133 

16.3 

32.6 

48.9 

65.2 

81.5 

97.8 

114.1 

130.4 

146.7 

1 12 

16.2 

32.4 

48.5 

64.8 

81.0 

97.2 

113.4 

129.6 

145.8 

151 

16.1 

32.2 

48.3  | 

64.4 

80.5  | 

96.6 

112.7 

128.8 

144.9 

TABLE  XI.  LOGARITHMS  OF  NUMBERS. 


179 


No.  270  L 431.]  [No.  290  L.  47G. 


N. 

0 

1 

2 

3 

4 

! | 
:_m 

6 

7 

8 

9 

Liff. 

270 

431304 

1525 

1685 

1846 

2007 

2167 

2328 

2488 

2649 

2809 

101 

1 

2909 

3130 

3290 

3450 

3010 

3770  | 

3930 

4090 

4249 

4409 

100 

2 

4509 

4729 

4888 

5048 

5207 

5307 

5526 

5085 

5844 

6004 

159 

3 

6103 

0322 

6481 

0040 

6799 

0957 

7116 

7276 

7433 

7592 

159 

4 

7751 

7909 

8007 

8220 

8384 

8542 

8701 

8859 

9017 

9175 

158 

5 

9333 

9491 

9048 

9800 

9904 

l. 

— 

— 

0122 

0279 

0437 

0594 

0752 

158 

6 

440909 

1006 

1224 

1381 

1538 

1095 

1852 

2009 

2106 

2323 

157 

7 

2480 

2037 

2793 

2950 

3106 

3203 

3419 

3576 

3732 

3889 

157 

8 

4045 

4201 

4357 

4513 

4009 

4825 

4981 

5137 

5293 

5449 

156 

9 

5004 

5700 

5915 

6071 

6226 

6382 

0537 

6092 

6848 

7003 

155 

280 

7158 

7313 

7468 

7623 

7778 

7933 

8088 

8242 

8397 

8552 

155 

1 

8706 

8861 

9015 

9170 

9324 

9478 

9033 

9787 

9941 

0095 

154 

2 

450249 

0403 

0557 

0711 

0865 

1018 

1172 

1326 

1479 

1633 

154 

3 

1786 

1940 

2093 

2247 

2400 

2553 

2706 

2859 

3012 

3165 

153 

4 

3318 

3471 

3624 

3777 

3930 

4082 

4235 

4387 

4540 

4692 

153 

5 

4845 

4997 

5150 

5302 

5454 

5600 

5758 

5910 

6002 

6214 

152 

6 

6306 

6518 

6070 

6821 

6973 

7125 

7276 

7428 

757'9 

7731 

152 

7 

7882 

8033 

8184 

8336 

8487 

8638 

87'89 

8940 

9091 

9242 

151 

8 

9392 

9543 

9694 

9845 

9995 

0146 

| 0296 

0447 

0597 

0748 

151 

9 

460898 

1048 

1198 

1348 

1499 

1649 

: 1799 

1948 

2098 

2248 

150 

290 

2398 

2548 

2697 

2847 

2997 

3146 

3296 

3445 

3594 

3744 

150 

1 

3893 

4042 

4191 

4340 

4490 

4039 

I 4788 

4936 

5085 

5234 

149 

2 

5383 

5532 

5680 

5829 

5977 

: 0126 

! 6274 

6423 

6571 

6719 

149 

3 

6868 

7016 

7164 

7312 

7460  1 

7608 

1 7756 

7904 

8052 

8200 

148 

4 

8347 

8495 

8643 

8790 

8938  ! 

9085 

9233 

9380 

9527 

9075 

148 

5 

9822 

9969 

0116 

0263 

0410 

0557 

0704 

0851 

0998 

1145 

147 

6 

471292 

1438 

1585 

1732 

1878 

2025 

2171 

2318 

2464 

2010 

146 

7 

2756 

2903 

3049 

3195 

3341 

3487 

3633 

3779 

3925 

4071 

146 

8 

4216 

4362 

4508 

4653 

4799 

4944 

5090 

5235 

5381 

5526 

146 

9 

5671 

5816 

5962 

6107 

6252 

6397 

i 6542 

6687 

6832 

6976 

145 

Proportional  Parts. 


Difif. 

1 

2 

3 

4 

5 

6 

7 

8 

9 

161 

16.1 

32.2 

48.3 

64.4 

80.5 

96.6 

112.7 

128.8 

144.9 

160 

! 16.0 

32.0 

48.0 

64.0 

80.0 

96.0 

112.0 

128.0 

144.0 

159 

i 15.9 

31.8 

47.7 

63.6 

79.5 

95.4 

111.3 

127.2 

143.1 

158 

! 15.8 

31.6 

47.4 

63.2 

79.0 

94.8 

110.6 

126.4 

142.2 

157 

15.7 

31.4 

47.1 

62.8 

78.5 

94.2 

109.9 

125.6 

141.3 

156 

15.6 

31.2 

46.8 

62.4 

78.0 

93.6 

109.2 

124.8 

140.4 

155 

15.5 

31.0 

46.5 

62.0 

77.5 

93.0 

108.5 

124.0 

139.5 

154 

15.4 

30.8 

46.2 

61.6 

77.0 

92.4 

107.8 

123.2 

138.6 

153 

15.3 

30.6 

45.9 

61.2 

76.5 

91.8 

107.1 

122.4 

137.7 

152 

15.2 

30.4 

45.6 

60.8 

76:0 

91.2 

106.4 

121.6 

136.8 

151 

15.1 

30.2 

45.3 

60.4 

75.5 

90.6 

105.7 

120.8 

135.9 

150 

15.0 

30.0 

45.0 

60.0 

75.0 

90.0 

105.0 

120.0 

135.0 

149 

14.9 

29.8 

44.7 

59.6 

74.5 

89.4 

104.3 

119.2 

134.1 

148 

14.8 

29.6 

44.4 

59.2 

74.0 

88.8 

103.6 

118.4 

133.2 

147 

14.7 

29.4 

44.1 

58.8 

73.5 

88.2 

102.9 

117.6 

132.3 

146 

14.6 

29.2 

43.8 

58.4 

73.0 

87.6 

102.2 

116.8 

131.4 

145 

14.5 

29.0 

43.5 

58.0 

72.5 

87.0 

101.5 

116.0 

130.5 

144 

14.4 

28.8 

43.2 

57.6 

72.0 

86.4 

100.8 

115.2 

129.6 

143 

14.3 

28.6 

42.9 

57.2 

71.5 

85.8 

100.1 

114.4 

128.7 

142 

14.2 

28.4 

42.6 

56.8 

71.0 

85  2 

99.4 

113.6 

127.8 

141 

14.1 

28.2 

42.3 

50.4 

70.5 

84.6 

98.7 

112.8 

126.9 

140 

14.0 

28.0 

42.0 

56.0 

70.0 

84.0 

98.0 

112.0 

126.0 

180 


TABLE  XI.  LOGARITHMS  OF  NUMBERS, 


No.  300  L.  477.]  [No.  339  L.  531. 


N. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

Dm. 

300 

477121 

7266 

7411 

7555 

7700 

7844 

7989 

8133  i 

8278  ! 

8422 

145 

1 

8566 

8711 

8855 

8999 

9143 

9287 

9431 

957 5 

9719  j 

9863 

144 

2 

480007 

0151 

0294 

0438 

0582 

0725 

0869 

1012 

1156 

1299 

144 

3 

1443 

1586 

1729 

1872 

2016 

2159 

2302 

2445 

2588 

2731 

143 

4 

2874 

3016 

3159 

3302 

3445 

3587 

3730 

3872 

4015 

4157 

143 

5 

4300 

4442 

4585 

4727 

4869 

5011 

5153 

5295 

5437 

5579 

142 

6 

5721 

5863 

6005 

6147 

6289 

6430 

6572 

6714 

6855 

6997 

142 

7 

7138 

7280 

7421 

7563 

7704 

7845 

7986 

8127 

8269 

8410 

141 

8 

8551 

8692 

8833 

8974 

9114 

9255 

9396 

9537 

9677 

9818 

141 

9 

9958 

0099 

0239 

0380 

0520 

0661 

0801 

0941 

1081 

1222 

140 

310 

491362 

1502 

1642 

1782 

1922 

2062 

2201 

2341 

2481 

2621 

140 

1 

2760 

2900 

3040 

3179 

3319 

3458 

3597 

8737 

3876 

4015 

139 

2 

4155 

4294 

4433 

4572 

4711 

4850 

4989 

5128 

5267 

5406 

139 

3 

5544 

5683 

5822 

5960 

6099 

6238 

6376 

6515 

6653 

6791 

139 

4 

6930 

7068 

7206 

7344 

7483 

7621 

7759 

7897 

8035 

8173 

138 

5 

8311 

8448 

8586 

8724 

8862 

8999 

9137 

9275 

9412 

9550 

138 

0 

9687 

9824 

9962 

0099 

0236 

0374 

0511 

0648 

0785 

0922 

137 

7. 

501059 

1196 

1333 

1470 

1607 

1744 

1880 

2017 

2154 

2291 

137 

8 

2427 

2564 

2700 

2837 

2973 

8109 

3246 

3382 

3518 

3655 

136 

9 

3791 

3927 

4063 

4199 

4335 

4471 

4607 

4743 

4878 

5014 

136 

320 

5150 

5286 

5421 

5557 

5693 

5828 

5964 

6099 

6234 

6370 

136 

1 

6505 

6640 

6776 

6911 

7046 

7181 

7816 

7451 

7586 

7721 

135 

2 

7856 

7991 

8126 

8260 

8395 

8530 

8664 

8759 

8934 

9068 

185 

3 

9203 

9337 

9471 

9606 

9740 

i 8874 

0009 

0143 

0277' 

0411 

134 

4 

510545 

0679 

0813 

0947 

1081 

1215 

1849 

1482 

1616 

1750 

134 

5 

1883 

2017 

2151 

2284 

2418 

2551 

2684 

2818 

2951 

3084 

133 

6 

3218 

3351 

3484 

3617 

3750 

3883 

4016 

4149 

4282 

4415 

133 

7 

4548 

4681 

4813 

4946 

5079 

5211 

5844 

5476 

5609 

5741 

133 

8 

5874 

6006 

6139 

6271 

6403 

6535 

6668 

6800 

6932 

7004 

132 

9 

7196 

7328 

7460 

7592 

7724 

7855 

7987 

8119 

8251 

8382 

132 

330 

8514 

8646 

8777 

8909 

9040 

9171 

9803 

9434 

9566 

9697 

131 

1 

9828 

9959 

0090 

0221 

0353 

i 0484 

0615 

0745 

0876 

1007 

131 

2 

521138 

1269 

1400 

1530 

1661 

1792 

1922 

2053 

2183 

2314 

131 

3 

2444 

2575 

2705 

2835 

29 66 

8096 

3226 

3356 

3486 

3616 

130 

4 

3746 

3876 

4006 

4136 

4266 

4396 

4526 

4656 

4785 

4915 

130 

5 

5045 

5174 

5304 

5434 

5563 

5693 

5822 

5951 

1 6081 

6210 

129 

6 

6339 

6469 

6598 

6727 

6856 

6985 

7114 

72^3 

; i O ( 2 

7501 

129 

, 7 

7630 

7759 

7888 

8016 

8145 

I 8274 

8402 

8531 

8660 

£788 

129 

8 

8917 

9045 

9174 

9302 

9430 

9559 

9687 

9815 

9943 

0072 

188 

9 

530200 

0328 

0456 

0584 

0712 

I 0840 

0968 

1096 

1223 

1351 

128 

Proportional  Parts. 


Diff. 

1 

2 

3 

4 

5 

6 

7 

8 

9 

139 

13.9 

27.8 

41.7 

55.6 

69.5 

83.4 

97.3 

111.2 

125.1 

138 

13.8 

27.6 

41.4 

55.2 

69.0 

82.8 

96.6 

110.4 

124.2 

137 

13.7 

27.4 

41.1 

54.8 

68.5 

82.2 

95.9 

109.6 

123.3 

136 

13.6 

27.2 

40.8 

54.4 

68.0 

81.6 

95.2 

108.8 

122.4 

135 

13.5 

27.0 

40.5 

54.0 

67.5 

81.0 

94.5 

108.0 

121.5 

134 

13.4 

26.8 

40.2 

53.6 

67.0 

£0.4 

93.8 

107.2 

120.6 

133 

13.3 

26.6 

39.9 

53.2 

66.5 

79.8 

93.1 

106.4 

119.7 

132 

13.2 

26.4 

39.6 

52.8 

66.0 

79.2 

92.4 

105.6 

118.8 

131 

13.1 

26.2  • 

89.3 

52.4 

65.5 

. 7'8.6 

91.7 

104.8 

117.9 

130 

13.0 

26.0 

39.0 

52.0 

65.0 

78.0 

91.0 

104.0 

117.0 

129 

12.9 

25.8 

38.7 

51.6 

64.5 

77.4 

90.3 

103.2 

116.1 

128 

12.8 

25.6 

38.4 

51.2 

64.0 

70.8 

89.6  | 

102.4 

115.2 

127 

12  7 

25.4 

33.1 

50.3 

C3. 5 

7 o.2 

Go . 9 | 

101.6 

114.3 

TABU?  XI.  LOGARITHMS  OF  NUMBERS. 


181 


No.  340  L.  531.] 

[No.  379  L.  579. 

N. 

0 

I 

2 

3 

4 

5 

6 

7 

8 

9 j 

Diff. 

340 

531479 

1607 

1734 

1862 

1990 

2117 

2245 

2372 

2500 

2627  ! 

128 

1 

2754 

2882 

3009 

3136 

3264 

3391 

3518  ; 

3645 

37  72 

3899 

127 

2 

4026 

4153 

4280 

4407 

4534 

4661 

4787' 

4914 

5041 

5167 

127 

3 

5294 

5121 

5547 

5674 

5800  | 

5927 

6053 

6180 

6306 

6432 

126 

4 

6685 

6811 

6937 

7063  ! 

7189 

7315  ! 

7441 

7567 

7693 

126 

5 

7819 

7945 

8071 

8197 

8322 

8448 

8574  ! 

8699 

8825 

8951 

126 

6 

9076 

9202 

9327 

9452 

9578  1 

97  03 

9829 

9954 



0079 

0204 

125 

7 

540329 

0455 

0580 

0705 

0830 

0955 

1080 

1205 

1330 

1454 

125 

8 

1579 

1704 

1829 

1953 

2078 

2203 

2327 

2452 

2576 

2701 

125 

9 

2825 

2950 

3074 

3199 

3323 

3447 

3571 

3696 

3820 

3944 

124 

350 

4068 

4192 

4316 

4440 

4564 

4688 

4812 

4936 

5060 

5183 

124 

1 

5307 

5431 

5555 

5678 

5802 

5925 

6049 

6172 

6296 

6419 

124 

2 

6543 

6666 

6789 

6913 

7036 

7159 

7282 

74U5 

7529 

7652 

123 

3 

7775 

7898 

8021 

8144 

8267 

8389 

8512 

8635 

8758 

8881 

123 

4 

9003 

9126 

9249 

9371 

9494 

9616 

9739 

9861 

9984 

— 

0106 

123 

5 

550228 

0351 

0473 

0595 

0717 

0840 

0962 

1084 

1206 

1328 

122 

6 

1450 

1572 

1694 

1816 

1938 

2060 

2181 

2303 

2425 

2547 

122 

2668 

2790 

2911 

3033 

3155 

3276 

3398 

3519 

3640 

3762 

121 

8 

3883 

4004 

4126 

4247 

4368 

4489 

4610 

4731 

4852 

4973 

121 

9 

5094 

5215 

5336 

5457 

5578 

5699 

5820 

5940 

6061 

6182 

121 

360 

6303 

6423 

6544 

6664 

6785 

6905 

7026 

7146 

7267 

7387 

120 

1 

7507 

7627 

7748 

7868 

7988 

8108 

8228 

8349 

8469 

8589 

120 

2 

8709 

QQA7 

8829 

8948 

9068 

9188 

9308 

9428 

9548 

9667 

9787 

120 

3 

0026 

0146 

0265 

0385 

0504 

0624 

0743 

0863 

0982 

119 

4 

561101 

1221 

1340 

1459 

1578 

1698 

1817 

1936 

2055 

2174 

119 

5 

2293 

2412 

2531 

2650 

2769 

2887 

3006 

3125 

3244 

3362 

119 

6 

3181 

3600 

3718 

3837 

3955 

4074 

4192 

4311 

4429 

4548 

119 

7 

4666 

4784 

4903 

5021 

5139 

5257 

5376 

5494 

5612 

5730 

118 

8 

5848 

5966 

6084 

6202 

6320 

6437 

6555 

6673 

6791 

6909 

118 

9 

7026 

7144 

7262 

7379 

7497 

7614 

7732 

7849 

7967 

8084 

118 

370 

8202 

8319 

8436 

8554 

8671 

8788 

8905 

9023 

9140 

9257 

117 

1 

9374 

9491 

96u8 

9725 

9842 

9959 

0076 

0193 

0309 

0426 

117 

2 

570543 

0660 

0776 

0893 

1010 

1126 

1243 

1359 

1476 

1592 

117 

3 

1709 

1825 

1942 

2058 

2174 

2291 

2407 

2523 

2639 

2755 

116 

4 

2872 

2988 

3104 

3220 

3336 

3452 

3568 

3684 

3800 

3915 

116 

5 

4031 

4147 

4263 

4379 

4494 

4610 

47'26 

4841 

4957 

5072 

116 

6 

5188 

5303 

5419 

5534 

5650 

5765 

5880 

5996 

6111 

6226 

115 

7 

6341 

6457 

6572 

6687 

6802 

6917 

7032  . 

7147 

7262 

7377 

115 

8 

7492 

7607 

7722 

7836 

7951 

8056 

8181 

8295 

8410 

8525 

115 

9 

8639 

.8754 

8868 

8983 

9097 

9212 

9326. 

9441 

9555 

9669 

114 

Proportional.  Parts. 


Diff.* 

1 

2 

3 

4 

5 

6 

7 

8 

9 

128 

12.8 

25.6 

38.4 

51.2 

64.0 

76.8 

89.6 

102.4 

115.2 

127 

12.7 

25.4 

38.1 

50.8 

63.5 

76.2 

88.9 

101.6 

114.3 

126 

12.6 

25.2 

37.8 

50.4 

63.0 

75.6 

88.2 

100.8 

113.4 

125 

12.5 

25.0 

37.5 

50.0 

62.5 

75.0 

87.5 

100.0 

112.5 

124 

12.4 

24.8 

37.2 

49.6 

62.0 

74.4 

86.8 

99.2 

111.6 

123 

12.3 

24.6 

36.9 

49.2 

61.5 

73.8 

86.1 

98.4 

110.7 

122 

12.2 

24.4 

36.6 

48.8 

61.0 

73.2 

85.4 

97.6 

109.8 

121 

12.1 

24.2 

36.3 

48.4 

60.5 

72.6 

84.7 

96.8 

108.9 

120 

12.0 

24.0 

36.0 

48.0 

60.0 

72.0 

84.0 

96.0 

108.0 

l2!!L 

11.9 

23.8 

35.7 

. 47.6 

59.5 

71.4 

83.3 

95.2 

107.1 

182 


TABLE  XL  LOGARITHMS  OF  NUMBERS, 


No.  380.  L.  579. J [No.  414  L.  617. 


N. 

0 

1 

2 

3 

4 

5 

6 

1 

8 

9 

Diff. 

380 

579784 

9898 

0012 

0126 

0241 

”0355*' 

0469' 

0583 

0697 

0811 

114 

1 

580925 

1039 

1153 

1267 

1381 

1495 

1608  I 

1722 

1836 

1950 

2 

2063 

2177 

2291 

2404 

2518 

2631 

2745 

2858 

29 72 

3085 

3 

3199 

3312 

3426 

3539 

3652 

3765 

3879 

3992 

4105 

4218 

4 

4331 

4444 

4557 

4670 

47’83 

4896 

5009 

5122 

5235 

5348 

113 

5 

5461 

5574 

5686 

5799 

5912 

6024 

6137 

6250 

6362 

6475 

6 

6587 

6700 

6812 

6925 

7037 

7149 

7262 

7374 

7486 

7599 

7 

7711 

7823 

7935 

8047 

8160 

! 8272 

8384 

8496 

8608 

8720 

112 

8 

9 

8832 

9950 

8944 

9056 

9167 

9279 

9391 

9503 

9615 

9726 

9838 

0061 

0173 

0284 

0396 

0507 

0619 

0730 

0842 

0953 

300 

591065 

1176 

1287 

1399 

1510 

1621 

1732 

1843 

1955 

2066 

1 

2177 

2288 

2399 

2510 

2621 

2732 

2843 

2954 

3064 

3175 

111 

2 

3286 

3397 

3508 

3618 

3729 

3840 

3950 

4061 

4171 

4282 

3 

4393 

4503 

4614 

4724 

4834 

4945 

5055 

5165 

5276 

. 5386 

4 

5496 

5606 

> 5717 

5827 

5937 

6047 

6157 

6267 

6377 

6487 

110 

5 

6597 

6707 

6817 

6927 

7037 

7146 

7256 

7366 

7476 

7586 

6 

7695 

7805 

7914 

8024 

8134 

8243 

8353 

8462 

8572 

8681 

7 

8 

8791 

9883 

8900 

9992 

9009 

9119 

9228 

9337 

9446 

9556 

9665 

9774 

109 

— 



0101 

0210 

0319 

0428 

0537 

0646 

0755 

0864 

9 

600973 

1082 

1191 

1299 

1408 

1517 

1625 

1734 

1843 

1951 

400 

2060 

2169 

2277 

•2386 

2494 

2603 

2711 

2819 

2928 

3036 

1 

3144 

3253 

3361 

3469 

3577 

3686 

3794 

3902 

4010 

4118 

108 

2 

4226 

4334 

4442 

4550 

4658 

4766 

4874 

4982 

5089 

5197 

3 

5305 

5413 

5521 

5628 

5736 

5844 

5951 

6059 

6166 

6274 

4 

6381 

6489 

6596 

6704 

6811 

6919 

7026 

7133 

7241 

7348 

5 

7455 

7562 

7669 

7777 

7884 

7991 

8098 

8205 

8312 

8419 

107 

6 

8526 

8633 

8740 

8847 

8954 

9061 

9167 

9274 

9381 

9488 

7 

9594 

9701 

9808 

9914 

0021 

1086 

0128 

1192 

0234 

1298 

0341 

1405 

0447 

1511 

0554 

1617 

8 

610660 

0767 

0873 

0979 

9 

1723 

1829 

1936 

2042 

2148 

2254 

2360 

2466 

2572 

2678 

106 

410 

2784 

2890 

2996 

3102 

3207 

3313 

3419 

3525 

3630 

3736 

1 

3842 

3947 

4053 

4159 

4264 

4370 

4475 

4581 

4686 

4792 

2 

4897 

5003 

5108 

5213 

5319 

5424 

5529 

5634 

5740 

5845 

3 

5950 

6055 

6160 

6265 

6370 

6476 

6581 

6686 

6790 

6895 

105 

4 

7000 

7105 

7210 

7315 

7420 

7525 

7629 

7734 

7839 

7943 

Proportional  Parts. 


Diff. 

1 

2 

3 

4 

5 

6 

7 

8 

9 

118 

11.8 

23.6 

35.4 

47.2 

59.0 

70.8 

82.6 

94.4 

106.2 

117 

11.7 

23.4 

35.1 

46.8 

58.5 

70.2 

81.9 

93.6 

105.3 

116 

11.6 

23.2 

34.8 

46.4 

58.0 

69.6 

81.2 

92.8 

104.4 

115 

11.5 

23.0 

34.5 

46.0 

57.5 

69.0 

80.5 

92.0 

103.5 

114 

11.4 

22.8 

34.2 

45.6 

57.0 

68.4 

79.8 

91.2 

102.6 

113 

11.3 

22.6 

33.9 

45.2 

56.5 

67.8 

79.1 

90.4 

101.7 

112 

11.2 

22.4 

33.6 

44.8 

56.0 

67.2 

78.4 

89.6 

100.8 

111 

11.1 

22.2 

33.3 

44.4 

55.5 

66.6 

77.7 

88.8 

99.9 

110 

11.0 

22.0 

33.0 

44.0 

55.0 

66.0 

77.0 

88.0 

99.0 

109 

10.9 

21.8 

32.7 

43.6 

54.5 

65.4 

76.3 

87.2 

98.1 

108 

10.8 

21.6 

32.4 

43.2. 

54.0 

64.8 

75.6 

86.4 

97.2 

107 

10.7 

21.4 

32.1 

42.8 

53.5 

64.2 

74.9 

85.6 

96.3 

106 

10.6 

21.2 

31.8 

42.4 

53.0 

63.6 

74.2 

84.8 

95.4 

105 

10.5 

21.0 

31.5 

42.0 

52.5 

63.0 

73.5 

84.0 

94.5 

105 

10.5 

21.0 

31.5 

42.0 

52.5 

63.0 

73.5 

84.0 

94.5 

104 

10.4 

20.8 

31.2 

41.6 

52.0 

62.4 

72.8 

83.2 

93.6 

TABLE  XI.  LOGARITHMS  OF  NUMBERS, 


183 


No.  415  L.  G18.]  [No.  459  L.  602 


N. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

Diff. 

415 

C18048 

8153 

8257 

8362 

8466 

8571 

8676 

8780 

8884 

8989 

105 

0 

9093 

9198 

9302 

9406 

9511 

9615 

9719 

9824 

9928 

0032 

620136 

0240 

0344 

0448 

0552 

0656 

0760 

0864 

0968 

1072 

104 

8 

1176 

1280 

1384 

1488 

1592 

1695 

1799 

1903 

2007 

2110 

9 

2214 

2318 

2421 

2525 

2628 

2732 

2835 

2939 

3042 

3146 

420 

3249 

3353 

3456 

3559 

3663 

3766 

3869 

3973 

4076 

4179 

1 

4282 

4385 

4488 

4591 

4695 

4798 

4901 

5004 

5107 

5210 

103 

2 

5312 

5415 

5518 

5621 

5724 

5827 

5929 

6032 

6135 

6238 

3 

G340 

6443 

6546 

6648 

6751 

6853 

6956 

7058 

7161 

7263 

4 

7366 

7468 

7571 

7673 

7775 

7878 

7980 

8082 

8185 

8287 

5 

8389 

8491 

8593 

8695 

8797 

8900 

9002 

9104 

9206 

9308 

102 

6 

9410 

9512 

9613 

9715 

9817 

9919 

1 0021 

0123 

0224 

0326 

630428 

0530 

0631 

0733 

0835 

0936 

10S8 

1139 

1241 

1342 

8 

1444 

1545 

1647 

1748 

1849 

1951 

1 2052 

2153 

2255 

2356 

9 

' 2457 

2559 

2660 

2761 

2862 

2963 

3064 

3165 

3266 

3367 

430 

3468 

3569 

3670 

3771 

3872 

3973 

4074 

4175 

4276 

4376 

101 

1 

4477 

4578 

4679 

4779 

4880 

4981 

5081 

5182 

5283 

5383 

2 

5484 

5584 

5685 

5785 

5886 

5986 

! 6087 

6187 

6287 

6388 

3 

6488 

6588 

6688 

6789 

6889 

• 6989 

7089 

7189 

7290 

7390 

4 

7490 

7590 

7690 

7790 

7890 

7990 

8090 

8190 

8290 

8889 

100 

5 

8489 

8589 

8689 

8789 

8888 

8988 

9088 

9188 

9287 

9387 

6 

9486 

9586 

9686 

9785 

9885 

9984 

— 

„ 

0084 

0183 

0283 

0382 

7 

640481 

0581 

0680 

0779 

0879 

0978 

1077 

1177 

1276 

1375 

8 

1474 

1573 

1672 

1771 

1871 

1970 

2069 

2168 

2267 

2366 

9 

2465 

2563 

2662 

2761 

2860 

2959 

3058 

3156 

3255 

3354 

99 

440 

3453 

3551 

3650 

3749 

3847 

3946 

4044 

4143 

4242 

4340 

1 

4439 

4537 

4636 

4734 

48S2 

4931 

5029 

5127 

5226 

5324 

2 

5422 

5521 

5619 

5717 

5815 

5913 

6011 

6110 

6208 

6306 

3 

6404 

6502 

6600 

6698 

6796 

6894 

6992 

7089 

7187 

7285 

98 

4 

7383 

7481 

7579 

7676 

7774 

7872 

7969 

8067 

8165 

8262 

5 

8360 

8458 

8555 

8653 

8750 

8848 

. 8945 

9043 

9140 

9237 

6 

9335 

9432 

9530 

9627 

9724 

9821 

9919 



— 

0016 

0113 

0210 

7 

650308 

0405 

0502 

0599 

0696 

0793 

0890 

0987 

1084 

1181 

8 

1278 

1375 

1472 

1569 

1666 

1762 

1859 

1956 

2053 

2150 

97 

9 

2246 

2343 

2440 

2536 

2633 

2730 

2826 

2923 

3019 

3116 

450 

3213 

3309 

3405 

3502 

3598 

3695 

3791 

3888 

3984 

4080 

1 

4177 

4273 

4369 

4465 

4562 

4658 

4754 

4850 

4946 

5042 

2 

5138 

5235 

5331 

5427 

5523 

5619 

5715 

5810 

5906 

6002 

96 

3 

6098 

6194 

6290 

6386 

6482 

6577 

6673 

6769 

6864 

6960 

4 

7056 

,7152 

7247 

7343 

7438 

7534 

7629 

7725 

7820 

7916 

5 

8011 

8107 

8202 

8298 

8393 

8488 

8584 

8679 

8774 

8870 

6 

8965 

9060 

9155 

9250 

9346 

9441 

9536 

9631 

9726 

9821 

7 

9916 

0011 

0106 

0201 

0296 

0391 

0486 

0581 

0676 

0771 

95 

8 

660865 

0960 

1055 

1150 

1245 

1339 

1434 

1529 

1623 

1718 

9 

1813 

1907 

2002 

2096 

2191 

•2286 

2380 

2475 

2569 

2663 

Proportional  Parts. 


Diff. 

1 

2 

3 

4 

5 

6 

7 

8 

9 

105 

10.5 

21.0 

31.5 

42.0 

52.5 

63.0 

73.5 

84.0 

94.5 

104 

10.4 

20.8 

31.2 

41.6 

52.0 

62.4 

72  8 

83.2 

93.6 

103 

10.3 

20.6 

30.9 

41.2 

51.5 

61.8 

72  1 

82.4 

92.7 

102 

10.2 

20.4 

30.6 

40.8 

51.0 

61.2 

71  4 

81.6 

91.8 

101 

10.1 

20.2 

30.3 

40.4 

50.5 

60.6 

70  7 

80.8 

90.9 

100 

10.0 

20.0 

30.0 

40.0 

50.0 

60.0 

70  0 

80.0 

90.0 

99 

9.9 

19.8 

29.7 

39.6 

49.5 

59.4 

69.3 

79.2 

89.1 

184 


TABLE  XI.  LOGARITHMS  OF  NUMBERS, 


No.  460  L.  662.] 


LNo.  499  L.  698. 


N 

0 

1 

2 

8 

4 

5 

6 

7 

8 

9 

Diff. 

460 

662758 

2852 

2947 

3041 

3135 

3230 

3324 

3418 

3512 

3607 

1 

3701 

3795 

3889 

3983 

4078 

4172 

4266 

4360 

4454 

4548 

2 

4642 

4736 

4830 

4924 

5018 

5112 

5206 

5299 

5393 

5487 

94 

3 

5581 

5675 

5769 

5862 

5956 

6050 

6143 

6237 

6331 

6424 

4 

6518 

6612 

6705 

6799 

6892 

6986 

7079 

7173 

7266 

7360 

5 

7453 

7546 

7640 

7733 

7826 

7920 

8013 

8106 

8199 

8293 

6 

8386 

8479 

8572 

8665 

8759 

8852 

8945 

9038 

9131 

9224 

7 

9317 

9410 

9503 

9596 

9689 

9782 

9875 

9967 

0060 

0153 

93 

8 

670246 

0339 

0431 

0524 

0617 

0710 

0802 

0895 

0988 

1080 

9 

1173 

1265 

1358 

1451 

1543 

1636 

1728 

1821 

1913 

2005 

470 

2098 

2190 

2283 

2375 

2467 

2560 

2652 

2744 

2836 

2929 

1 

3021 

3113 

3205 

3297 

3390 

3482 

3574 

3666 

3758 

3850 

2 

3942 

4034 

4126 

4218 

4310 

4402 

4494 

4586 

i 4677 

4769 

92 

3 

4861 

4953 

5045 

5137 

5228 

5320 

5412 

5503 

5595 

5687 

4 

5778 

5870 

5962 

6053 

6145 

6236 

6328 

6419 

6511 

6602 

5 

6694 

6785 

6876 

6968 

7059 

7151 

7242 

7333 

7424 

7516 

6 

7607 

7698 

7789 

7881 

7972 

8063 

8154 

8245 

8336 

8427 

7 

8518 

8609 

8700 

8791 

8882 

! 8973 

9064 

9155 

9246 

9337 

91 

8 

9428 

9519 

9610 

9700 

9791 

9882 

9973 

0063 

0154 

0245 

9 

680336 

0426 

0517 

0607 

0698- 

0789 

0879 

0970 

1060 

1151 

480 

1241 

1332 

1422 

1513 

1603 

1693 

1784 

1874 

1964 

2055 

1 

2145 

2235 

2326 

2416 

2506 

2596 

2686 

2777 

2867 

2957 

2 

3047 

3137 

3227 

3317 

3407 

3497 

3587 

3677 

3767 

3857 

90 

3 

3947 

4037 

4127 

4217_ 

4307 

4396 

4486 

4576 

4666 

4756 

4 

4845 

4935 

5025 

'5114 

5204 

5294 

5383 

5473 

5563 

5652 

5 

5742 

5831 

5921 

6010 

6100 

6189 

6279 

6368 

6458 

6547 

6 

6636 

6726 

6815 

6904 

6994 

7083 

7172 

7261 

7351 

7440 

7 

7529 

7618 

7707 

7796 

7886 

7975 

8064 

8153 

8242 

8331 

89 

8 

8420 

8509 

8598 

8687 

8776 

8865 

8953 

9042 

9131 

9220 

9 

9309 

9398 

9486 

9575 

9664  ; 

9753 

9841 

9930 

0019 

0107 

490 

690196 

0285 

0373 

0462 

0550 

0639 

0728 

0816 

0905 

0993 

1 

1081 

1170 

1258 

1347 

1435  i 

1524 

1612 

1700 

1789 

1877 

2 

1965 

2053 

2142 

2230 

2318 

2406 

2494 

2583 

2671 

2759 

3 

2847 

2935 

3023 

3111 

3199  ! 

3287 

3375 

3463 

3551 

3639 

88 

4 

3727 

3815 

3903 

3991 

4078  j 

4166 

4254 

4342 

4430 

4517 

5 

4605 

4693 

4731 

4868 

4956  ! 

5044 

5131 

5219 

5307 

5394 

6 

5482 

5569 

5657 

5744 

5832  j 

5919 

6007 

6094 

6182 

6269 

7 

6356 

6444 

6531 

6618 

6706  ; 

6793 

6880 

6968 

7055 

7142 

8 

7229 

7317 

7404 

7491 

7578  j 

7665 

7752 

7839 

7926 

8014 

87 

9 

8100 

8188 

8275 

8362 

8449 

1 

8535 

8622 

8709 

8796 

8883 

Proportional  Parts. 


Diff. 

1 

2 

3 

4 

5 

6 

7 

8 

9 

98 

9.8 

19.6 

29.4 

39.2 

49.0 

58.8 

68.6 

78.4 

88.2 

97 

9.7 

19.4 

29.1 

38.8 

48.5 

58.2 

67.9 

77.6 

87.3 

96 

9.6 

19.2 

28.8 

38.4 

48.0 

57.6 

67.2 

76.8 

86.4 

95 

9.5 

19.0 

28.5 

38.0 

47.5 

57.0 

66.5 

76.0 

85.5 

94 

9.4 

18.8 

28.2 

37.6 

47.0 

56.4 

65.8 

75.2 

84.6 

93 

9.3 

18.6 

27.9  | 

37.2 

46.5 

55.8 

65.1 

74.4 

83.7 

92 

9.2 

18.4 

27.6 

36.8 

46.0 

55.2 

64.4 

73.6 

82.8 

91 

9.1 

18.2 

27.3 

36.4 

45.5 

54.6 

.63.7 

72.8 

81.9 

90 

9.0 

18.0 

27.0 

36.0 

45.0 

54.0 

63.0 

72.0 

81.0 

89 

8.9 

17.8 

26.7 

35.6 

44.5 

53.4 

62.3 

71.2 

80.1 

88 

8.8 

17.6 

26.4  1 

35.2 

44.0 

52.8 

61.6 

70.4 

79.2 

87 

8.7 

17.4 

26.1  I 

34.B 

43.5 

52.2 

'60.9 

69.6 

78.3 

86 

8.6 

17.2 

25.8  | 

34.4 

43.0 

51.6 

60.2 

68.8 

77.4 

TABLE  XI.  LOGARITHMS  OF  NUMBERS. 


185 


_ — — ■ — -1 

No.  500  L.  698.]  [No.  544  L.  736. 


N. 

0 

1 

2 

3 

4 i 

5 

6 

7 

8 

9 

Diff. 

500 

1 

698970 

9838 

9057 

9924 

9144 

9231 

9317 

9404 

9491 

9578 

9664 

9751 

0011 

0098 

0184 

0271 

0358 

0444 

0531 

0617 

2 

700704 

0790 

0877 

0963 

1050 

1136 

1222 

1309 

1395 

1482 

3 

1568 

1654 

1741 

1827 

1913 

1999 

2086  ■ 

2172 

2258 

2341 

4 

2431 

2517 

2603 

2689 

2775 

2861 

2947 

3033 

3119 

3205 

5 

3291 

3377 

3463 

3549 

3635 

3721 

3807 

3893 

3979 

4065 

86 

6 

4151 

4236 

4322 

4408 

4494 

4579 

4665 

4751 

4837 

4922 

7 

5008 

5094 

5179 

5265 

5350 

5436 

5522 

5607 

5693 

5778 

8 

5864 

5949 

6035 

6120 

6206 

6291 

6376 

6462 

6547 

6632 

9 

6718 

6803 

6888 

6974 

7059  : 

7144 

7’229 

7315 

7400 

7485 

510 

7570 

7655 

7740 

7826 

7911 

7996 

8081 

8166 

8251 

8336 

85 

1 

8421 

8506 

8591 

8676 

8761 

8846 

8931 

9015 

9100 

9185 

2 

9270 

9355 

9440 

9524 

9609 

9694 

9779 

9863 

9948 

0033 

3 

710117 

0202 

0287 

0371 

0456 

0540 

0625 

0710 

0794 

0879 

4 

0963 

1048 

1132 

1217 

1301 

1385 

1470 

1554 

1639 

1723 

5 

1807 

1892 

1976 

2060 

2144 

2229 

2313 

2397 

2481 

2566 

6 

2650 

2734 

2818 

2902 

2986 

3070 

3154 

3238 

3323 

3407 

84 

7 

3491 

3575 

3659 

3742 

3826 

3910 

3994 

4078 

4162 

4246 

8 

4330 

4414 

4497 

4581 

4665 

4749 

4833 

4916 

5000 

5084 

9 

5167 

5251 

5335 

5418 

5502 

5586 

5669 

5753 

5836 

5920 

520 

6003 

6087 

6170 

6254 

6337 

6421 

6504 

6588 

6671 

6754 

1 

6838 

6921 

7004 

7088 

7171 

7254 

7338 

7421 

7504 

7587 

2 

7671 

7754 

7837 

7920 

8003 

8086 

8169 

8253 

8336 

8419 

83 

3 

8502 

8585 

8668 

8751 

8834 

8917 

9000 

9083 

9165 

9248 

4 

9331 

9414 

9497 

9580 

9663 

9745 

9828 

9911 

9994 

0077 

5 

720159 

0242 

0325 

0407 

0490 

0573 

0055 

0738 

0821 

0903 

6 

0986 

1068 

1151 

1233 

1316 

1398 

1481 

1563 

1646 

1728 

7 

1811 

1893 

1975 

2058 

2140 

2222 

2305 

2387 

2469 

2552 

8 

2634 

2716 

2798 

2881 

2963 

3045 

3127 

3209 

3291 

3374 

9 

3456 

3538 

3620 

3702 

3784 

3866 

3948 

4030 

4112 

4194 

82 

530 

4276 

4358 

4440 

4522 

4604 

4685 

4767 

4849 

49al 

5013 

1 

5095 

5176 

5258 

5340 

5422 

5503 

5585 

5667 

5748 

5830 

2 

5912 

5993 

6075 

6156 

6238 

6320 

6401 

6483 

6564 

6646 

3 

6727 

6809 

6890' 

6972 

7053 

7134 

7216 

7297 

7379 

7460 

4 

7541 

7623 

7704 

7785 

7866 

7948 

8029 

8110 

8191 

8273 

5 

8354 

8435 

8516 

8597 

8678 

8759 

8841 

8922 

9003 

9084 

6 

9165 

9246 

9327 

9408 

9489 

9570 

9651 

97’32 

9813 

9893 

81 

7 

9974 

0055 

0136 

0217 

0298 

0378 

0459 

0540 

0621 

0702 

8 

730782 

0863 

0944 

1024 

1105 

1186 

1266 

1347 

1428 

1508 

9 

1589 

1669 

1750 

1830 

1911 

1991 

2072 

2152 

2233 

2313 

540 

2394 

2474 

2555 

2635 

2715 

2796 

2876 

2956 

3037 

3117 

1 

3197 

3278 

3358 

3438 

| 3518 

3598 

3679 

3759 

3839 

3919 

2 

3999 

4079 

4160 

4240 

[ 4320 

4400 

4480 

4560 

4640 

4720 

80 

3 

4800 

4880 

4960 

5040 

| 5120 

5200 

5279 

5359 

5439 

5519 

4 

5599 

5679 

5759 

5838 

1 5918 

5998 

6078 

6157 

6237 

6317 

Proportional  Parts. 


Diff. 

1 

2 

3 

4 

5 

6 

7 

8 

9 

87 

8.7 

17.4 

26.1 

34.8 

43.5 

52.2 

60.9 

69.6 

78.3 

86 

8.6 

17.2 

*25.8 

34.4 

43.0 

51.6 

60.2 

68.8 

77.4 

85 

8.5 

17.0 

25.5 

34.0 

42.5 

51.0 

59.5 

68.0 

76.5 

84 

8.4 

16.8 

25.2 

33.6 

42.0  I 

50.4 

58.8 

67.2 

75.6 

186 


TABLE  XI.  LOGARITHMS  OP  NUMBERS, 


No.  545  L.  736.]  [No.  584  L.  767. 


N. 

0 

1 

2 

3 

4 

6 

6 

7 

8 ! 

9 

Diff. 

545 

736397 

6476 

6556 

6635 

6715 

6795 

6874 

6954 

7034 

7113  i 

6 

7193 

7272 

7352 

7431 

7511 

7590 

7670 

7749 

7829 

7908 

7 

7987 

8067 

8146 

8225 

8305  ! 

8384 

8463 

8543 

8622 

8701 

8 

8781 

8860 

8939 

9018 

9097 

9177 

9256 

9335 

9414 

9493 

9 

9572 

9651 

9731 

9810 

9889 

9968 

| 

0047 

0126 

0205 

0284 

79 

550 

740363 

0442 

0521 

0600 

0678 

*0757 

0836 

0915 

0994 

1073 

1 

1152 

1230 

1309 

1388 

1467 

1546 

1624 

1703 

1782 

1860 

2 

1939 

2018 

2096 

2175 

2254 

2332 

2411 

2489 

2568 

2647 

3 

2725 

2804 

2882 

2961 

3039 

3118 

3196 

3275 

3353 

3431 

4 

3510 

3588 

3667 

3745 

3823 

3902 

3980 

4058 

4136 

4215 

5 

4293 

4371 

4449 

4528 

4606 

4684 

4762 

4840 

4919 

4997 

6 

5075 

5153 

5231 

5309 

5387 

5465 

5543 

5621 

5699 

5777 

78 

4 

5855 

5933 

6011 

6089 

6167 

1 6245 

6323 

6401 

6479 

6556 

8 

6634 

6712 

6790 

6868 

6945 

1 7023  ! 

7101 

7179 

7256 

7334 

9 

7412 

7489 

7567 

7645 

7722 

7800 

7878 

7955 

8033 

8110 

560 

8188 

8266 

8343 

8421 

8498 

8576 

8653 

8731 

8808 

8885 

1' 

8963 

9040 

9118 

9195 

9272 

9350 

9427 

9504 

9582 

9659 

o 

9736 

9814 

9891 

9968 

0045 

0123 

0200 

0277 

0354 

0431 

3 

750508 

0586 

0663 

0740 

0817 

C894 

0971 

1048 

1125 

1202 

4 

1279 

1356 

1433 

1510 

1587 

1604 

1741 

1818 

1895 

1972 

77 

5 

2048 

2125 

2202 

2279 

2356 

2433 

2509 

2586 

2663 

2740 

i a 

6 

2816 

2893 

2970 

3047 

3123 

3200 

3277 

3353 

3430 

3506 

7 

3583 

3660 

3736 

3813 

3889 

3966 

4042 

4119 

4195 

4272 

8 

4348 

4425 

4501 

4578 

4654 

4730 

; 4807 

4883 

4960 

5036 

9 

5112 

5189 

5265 

5341 

5417 

5494 

5570 

5646 

5722 

5799 

570 

5875 

5951 

6027 

6103 

6180 

1 6256 

6332 

6408 

6484 

6560 

1 

6636 

6712 

6788 

6864 

6940 

7016 

7092 

7168 

7244 

7320 

76 

2 

7396 

7472 

7548 

7624 

7700 

7775 

! 7851 

7927 

8003 

8079 

3 

8155 

8230 

8306 

8382 

8458 

8533 

j 8609 

8685 

8761 

8836 

4 

8912 

8988 

9063 

9139 

9214 

9290 

9366 

9441 

9517 

9592 

5 

9668 

9743 

9819 

9894 

9970 

0045 

1 0121 

0196 

0272 

0347 

6 

760422 

0498 

0573 

0649 

0724 

0799 

i 0875 

0950 

1025 

1101 

7 

1176 

1251 

1326 

1402 

1477 

1552 

1627 

1702 

1778 

1853 

8 

1928 

2003 

2078- 

2153 

2228 

2303 

2378 

2453 

2529 

2604 

75 

9 

2679 

2754 

2829 

2904 

2978 

i 3053 

3128 

3203 

3278 

3353 

580 

3428 

3503 

3578 

3653 

3727 

! 3802 

3877 

3952 

4027 

4101 

1 

4176 

4251 

4326 

4400 

4475 

4550 

4624 

4699 

4774 

4848 

2 

4923 

4998 

5072 

5147 

5221 

| 5296 

i 5370 

5445 

5520 

5594 

3 

5669 

5743 

5818 

5892 

5966 

I 6041 

6115 

6190 

6264 

6338 

4 

6413 

6487 

. 6562 

6636 

6710 

I 6785 

! 6859 

6933 

7007 

7082 

Proportional  Parts. 


Diff. 

1 

2 

3 

4 

5 

6 

7 

8 

9 

83 

8.3 

16.6 

24.9 

33.2 

41.5 

49.8 

58.1 

66.4 

74.7 

82 

8.2 

16.4 

24.6 

32.8 

41.0 

49.2 

57.4 

65.6 

73.8 

81 

8.1 

16.2 

24.3 

32.4 

40.5 

48.6 

56.7 

64.8 

72  9 

80 

8.0 

16.0 

24.0 

32.0 

40.0 

48.0 

56.0 

64.0 

72.0 

79 

7.9 

15.8 

23.7 

31.6 

39.5 

47.4 

55.3 

63.2 

71.1 

78 

7.8 

15.6 

23.4 

31.2 

39.0 

46.8 

54.6 

62.4 

70.2 

77 

7.7 

15.4 

23.1 

30.8 

38.5 

46.2 

53.9 

61.6 

69.3 

76 

7.6 

15.2 

22.8 

30.4 

38.0 

45 . 6 

53.2 

60.8 

68.4 

75 

7.5 

15.0 

22.5 

30.0 

37.5 

45.0 

52.5 

60.0 

67.5 

74 

7.4 

14.8 

22.2 

29.6 

37.0 

44.4 

51.8 

59.2 

66.6 

TABLE  XI.  LOGARITHMS  OF  NUMBERS. 


18? 


No.  585  L.  767.]  [No.  629  L.  799. 


N. 

0 

1 

2 

3 

4 

! 5 

6 

7 

8 

9 

Diff. 

585 

767156 

7230 

7304 

7379 

7453 

7527 

7601 

7675 

7749 

7823 

6 

7898 

7972 

8046 

8120 

8194 

8268 

8342 

! 8416 

j 8490 

; 8564 

74 

7 

8638 

8712 

8786 

8860 

8934 

9008 

9082 

9156 

! 9230 

9303 

8 

9377 

9451 

9525 

9599 

9673 

9746 

9820 

9894 

9968 

— 

0042 

9 

770115 

0189 

0263 

0336 

0410 

0484 

0557 

0631 

j 0705 

0778 

590 

0852 

0926 

0999 

1073 

1146 

1220 

1293 

1367 

1440 

1514 

1 

1587 

1661 

1734 

1808 

1881 

1955 

2028 

2102 

! 2175 

2248 

2 

2322 

2395 

2468 

2542 

2615 

2688 

: 2762 

2835 

2908 

2981 

a 

3055 

3128 

3201 

3274 

3348 

3421 

3494 

3567 

! 3640 

3713 

4 

3786 

3860 

3933 

4006 

4079  ; 

4152 

4225 

4298 

1 4371 

4444 

73 

5 

4517 

4590 

4663 

4736 

4809  i 

4882 

! 4955 

5028 

5100 

5173 

6 

5246 

5319 

5392 

5465 

5538 

5610 

! 5683 

5756 

5829 

5902 

7 

5974 

6047 

6120 

6193 

6265 

6338 

6411 

6483 

6556 

6629 

8 

6701 

6774 

6846 

6919 

6992 

7064 

| 7137 

7209 

7282 

7354 

9 

7427 

7499 

7572 

7644 

7717 

7789 

| 7862 

7934 

8006 

8079 

600 

8151 

8224 

8296 

8368 

8441 

8513 

! 8585 

8658 

8730 

8802 

1 

8874 

8947 

9019 

9091 

9163 

9236 

| 9308 

1 9380 

9452 

9524 

2 

9596 

9669 

9741 

9813 

9885 

9957 

0029 

0101 

0173 

no  < x 

3 

780317 

0389 

0461 

0533 

0605 

0677 

0749 

0821 

0893 

0965 

(X 

4 

1037 

1109 

1181 

1253 

1324 

1396 

1468 

1540 

1612 

1684 

5 

1755 

1827 

1899 

1971 

2042 

S 2114 

2186 

2258 

2329 

2401 

6 

2473 

2544 

2616 

2688 

2759 

2831 

2902 

2974 

3046 

3117 

7 

3189 

3260 

3332 

3403 

3475 

3546 

3618 

3689 

3761 

3832 

8 

3904 

39<o 

4046 

4118 

4189 

4261 

4332 

4403 

4475 

4546 

9 

4617 

4689 

4760 

4831 

4902 

4974 

5045 

5116 

5187 

5259 

610 

5330 

5401 

5472 

5543 

5615 

5686 

5757 

5828 

5899 

5970 

1 

6041 

6112 

6183 

6254 

6325 

6396 

6467 

6538 

6609 

6680 

71 

2 

6751 

6822 

6893 

6964 

7035 

7U6 

! 7177 

7248 

7319 

7390 

3 

7460 

7531 

7602 

7673 

7744 

78x5 

‘ 7885 

7956 

8027 

8098 

4 

8168 

8239 

8310 

8381 

8451 

8522 

8593 

8663 

8734 

8804 

5 

8875 

8946 

9016 

9087 

9157 

9228 

9299 

9369 

9440 

9510 

6 

9581 

9651 

9722 

9792 

9863 

9933 

: 0004 

0074 

0144 

0215 

7 

790285 

0356 

0426 

0496 

0567 

0637 

0707 

0778 

0848 

0918 

8 

0988 

1059 

1129 

1199 

1269 

1340 

1410 

1480 

1550 

1620 

9 

1691 

1761 

1831 

1901 

1971 

2041 

2111 

2181 

2252 

2322 

620 

2392 

2462 

2532 

2602 

2672 

2742 

2812 

2882 

2952 

3022 

70 

1 

3092 

3162 

3231 

3301 

3371 

3441 

3511 

3581 

3651 

3721 

2 

3790 

3860 

3930 

4000 

4070 

4139 

4209 

4279 

4349 

4418 

3 

4488 

4558 

4627 

4697 

4767 

4836 

4906 

4976 

5045 

5115 

4 

5185 

5254 

5324 

5393 

5463 

5532 

5602 

5672  ! 

5741 

5811 

5 

5880. 

5949 

6019 

6088 

6158 

6227 

6297 

6366 

6436 

6505 

6 

6574' 

6644 

6713 

6782 

6852 

6921 

6990 

7060 

7129 

7198 

7 

7268 

7337 

7406 

7475 

7545 

7614 

7683 

7752  ! 

7821 

7890 

8 

7960 

8029 

8098 

8167 

8236  ! 

8305 

8374 

8443 

8513 

8582 

9 

8651 

8720 

8789 

8858 

8927  j 

8996 

9065 

91&4 

9203 

9272 

69 

Proportional  Parts. 


Diff. 

1 

2 

3 

4 

5 

6 

7 

8 

9 ’ 

75 

7.5 

15.0 

22.5 

30.0 

37.5 

45.0 

52.5 

60.0 

67.5 

74 

7.4 

14.8 

22.2 

1 29.6 

37.0 

44.4 

51.8 

59.2 

66.6 

73 

7.3 

14.6 

21.9 

1 29.2 

36.5 

43.8 

51.1 

58.4 

65.7 

72 

7.2 

14.4 

21.6 

28.8 

36.0 

43.2 

50.4 

57.6 

64.8 

71 

7.1 

14.2 

21.3 

28.4 

3o . 5 

42.6 

49.7 

56.8 

63.9 

70 

7.0 

14.0 

21.0 

28.0 

35.0 

42.0 

49.0 

56.0 

63.0 

69 

6.9 

13.8 

20.7 

| 27.6 

34.5 

41.4 

48.3 

55.2 

62.1 

188 


TABLE  XI.  LOGARITHMS  OF  NUMBERS, 


No.  630  L.  799.]  [No.  674  L.  829. 


N. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

9 

Diff. 

630 

799341 

9409 

947'8 

9547 

9616 

9685 

9754 

9823 

9892 

9961 

1 

800029 

0098 

0167 

0236 

0305 

0373 

0142 

0511 

0580 

0648 

2 

0717 

0786 

0854 

0923 

0992 

1061 

1129 

1198 

1266 

1335 

3 

1401 

1472 

1541 

1609 

1678 

1747 

1815 

1884 

1952 

2021 

4 

2089 

2158 

2226 

2295 

2363 

2432 

2500 

2568 

2637 

2705 

5 

2774 

2842 

2910 

2979 

3047 

3116 

3184 

3252 

3321 

3389 

6 

3457 

3525 

3594 

3662 

3730 

3798 

3867 

3935 

4003 

4071 

7 

4139 

4208 

4276 

4344 

4412 

4480 

4548 

4616 

4685 

4753 

8 

4821 

4889 

4957 

5025 

5093 

5161 

5229 

5297 

5365 

5433 

68 

9 

5501 

5569 

5637 

5705 

5773 

5841 

5908 

5976 

6044 

6112 

640 

806180 

6248 

6316 

6384 

6451 

6519 

6587 

6655 

6723 

6790 

1 

6858 

6926 

6994 

7061 

7129 

7197 

7264 

7332 

7400 

7467 

O 

7535 

7603 

7670 

7738 

7806 

7873 

7941 

8008  ' 

8076 

8143 

3 

8211 

8279 

8346 

8414 

8481 

8549 

8616 

8684 

8751 

8818 

4 

8886 

8953 

9021 

9088 

9156 

9223 

9290 

9358 

9425 

9492 

5 

9560 

9627 

9694 

9762 

9829 

9896 

9964 

0031 

0098 

0165 

6 

810233 

0300 

0367 

0434 

0501 

0569 

0636 

0703 

0770 

0837 

7 

0904 

0971 

1039 

1106 

1173 

1240 

1307 

1374 

1441 

1508 

67 

8 

1575 

1642 

1709 

1776 

1843 

1910 

1977 

2044 

2111 

2178 

9 

2245 

2312 

2379 

2445 

2512 

2579 

2646 

2713 

2780 

2847 

650 

2913 

2980 

3047 

3114 

3181 

3247 

3314 

3381 

3448 

3514 

1 

3581 

3648 

3714 

3781 

3848 

3914 

3981 

4048 

4114 

4181 

2 

4248 

4314 

4381 

4447 

4514 

4581 

4647 

4714 

4780 

4847 

3 

4913 

4980 

5046 

5113 

5179 

5246 

5312 

5378 

5445 

5511 

4 

5578 

5644 

5711 

5777 

5843 

5910 

5976 

6042 

6109 

6175 

5 

6241 

6308 

6374 

6440 

6506 

6573 

6639 

6705 

6771 

6838 

6 

6904 

6970 

7036 

7102 

7169 

7235 

7301 

7367 

7433 

7499 

7 

7565 

7631 

7698 

7764 

7830 

7896 

7962 

8028 

8094 

8160 

8 

8226 

8292 

8358 

8424 

8490 

8556 

8622 

8688 

8754 

8820 

Aft 

9 

8885 

8951 

9017 

9083 

9149 

9215 

9281 

9346 

9412 

9478 

OO 

660 

9544 

9610 

9676 

9741 

9807 

’ 9873 

9939 

0004 

0070 

0136 

1 

820201 

0267 

0333 

0399 

0464 

0530 

0595 

0661 

0727 

0792 

2 

0858 

0924 

0989 

1055 

1120 

1186 

1251 

1317 

1382 

1448 

3 

1514 

1579 

1645 

1730 

1775 

1841 

1906 

1972 

2037 

2103 

4 

2168 

2233 

2299 

2364 

2430 

2495 

2560 

2626 

2691 

2756 

5 

2822 

2887 

2952 

3018 

3083 

3148 

3213 

3279 

3344 

3409 

6 

3474 

3539 

3605 

3670 

3735 

3800 

3865 

3930 

3996 

4061 

7 

4126 

4191 

4256 

4321 

4386 

4451 

4516 

4581 

4046 

4711 

65 

8 

4776 

4841 

4906 

4971 

5036 

5101 

5166 

5231 

5296 

5361 

9 

5426 

5491 

5556 

5621 

5686 

5751 

5815 

5880 

5945 

6010 

670 

6075 

6140 

6204 

6269 

6334 

6399 

6464 

6528 

6593 

6658 

1 

6723 

6787 

6852 

6917 

6981 

'7046 

7111 

7175 

7240 

7305 

2 

7369 

7434 

7499 

7563 

7628 

7692 

7757 

7821 

7886 

7951 

3 

8015 

8080 

8144 

8209 

8273 

8338 

8402 

8467 

8531 

8595 

4 

§660 

8724 

87'89 

8853 

8918 

8982 

9046 

9111 

9175 

9239 

Proportional  Parts. 


Diff. 

1 

2 

3 

4 

5 

6 

7 

8 

9 

68 

6.8 

13.6 

20.4 

27.2 

34.0 

40.8 

47.6 

54.4 

61.2 

67 

6.7 

13.4 

20.1 

26.8 

33.5 

40.2 

46.9 

53.6 

60.3 

66 

6.6 

13.2 

19.8 

26.4 

33.0 

39.6 

46.2 

52.8 

59  4 

65 

6.5 

13.0 

19.5 

20.0 

32.5 

39.0 

45.5 

52.0 

58.5 

64 

6.4 

12.8 

19.2 

25.6 

32.0 

38.4 

44.8 

51.2 

57.6  j 

TABLE  XI.  LOGARITHMS  OF  NUMBERS. 


189 


No.  675  L.  829.]  [No.  719  L.  857. 


N. 

0 

1 

2 

3 

4 

6 

6 

7 

8 

9 

Diff 

675 

C 

• 829304 
9947 

9368 

9432 

9497 

9561 

9625 

9690 

9754 

9818 

9882 

0011 

0075 

0139 

0204 

0268 

0332 

0396 

0460 

0525 

7 

830589 

0653 

0717 

0781 

0845 

0909 

0973 

1037 

1102 

1166 

! 

8 

1230 

1294 

1358 

1422 

1486 

1550 

1614 

1678 

1742 

1806 

64 

9 

1870 

1934 

1998 

2062 

2126 

2189 

2253 

2317 

2381 

2445 

380 

2509 

2573 

2637 

2700 

2764 

2828 

2892 

2956 

3020 

3083 

1 

3147 

3211 

3275 

3338 

3402 

3466 

3530 

3593 

3657 

3721 

2 

3784 

3848 

3912 

3975 

4039 

4103 

4166 

4230 

4294 

4357 

3 

4421 

4484 

4548 

4611 

4675 

4739 

4802 

4866 

4929 

4993  ; 

4 

5056 

5120 

5183 

5247 

5881 

5310 

5373 

5437 

5500 

5564 

5627  ! 

5 

5691 

5754 

5817 

5944 

6007 

6071 

6134 

6197 

6261 

G 

6324 

6387 

6451 

6514 

6577 

6641 

6704 

6767 

6830 

6894 

7 

6957 

7020 

7083 

7146 

7210 

S 7273 

7336 

7399 

7462 

7525 

8 

7588 

7652 

7715 

7778 

7841 

7904 

7967 

8030 

8093 

8156 

63 

9 

8219 

8282 

8345 

8408 

8471 

8534 

8597 

8660 

8723 

8786 

690 

8849 

8912 

8975 

9038 

9101 

9164 

9227 

9289 

9352 

9415 

1 

9478 

9541 

9604 

9667 

9729 

9792 

9855 

9918 

9881 

0043 

0671 

2 

840106 

0169 

0232 

0294 

0357 

0120 

0482 

0545 

0608 

3 

0733 

0796 

0859 

0921 

0984 

1046 

1109 

1172 

1234 

1297 

4 

1359 

1422 

1485 

1547 

1610 

1672 

1735 

1797 

1860 

1922 

5 

1985 

2047 

2110 

2172 

2235 

2297 

2360 

2422 

2484 

2547 

6 

2609 

2672 

2734 

2796 

2859 

2921 

2983 

3046 

3108 

3170 

7 

3233 

3295 

3357 

3420 

3482 

3544 

3606 

3669 

3731 

37  93 

8 

3855 

3918 

3980 

4042 

4104 

4166 

4229 

4291 

4353 

4415 

9 

4477 

4539 

4601 

4664 

4726 

4788 

4850 

4912 

4974 

5036 

700 

5098 

5160 

5222 

5284 

5346 

5408 

5470 

5532 

5594 

5656 

62 

1 

5718 

5780 

5842 

5904 

5966 

6028 

6090 

6151 

6213 

6275 

2 

6337 

6399 

6461 

6523 

6585 

6646 

6708 

6770 

6832 

6894 

3 

6955 

7017 

7079 

7141 

7202 

7264 

7326 

7388 

7449 

7511 

4 

7573 

7634 

7696 

7758 

7819 

7881 

7943 

8004 

i 8066 

8128 

5 

8189 

8251 

8312 

8374 

8435 

8497 

8559 

8620 

8682 

874:3 

6 

8805 

8866 

8928 

8989 

9051 

9112 

9174 

9235 

9297  1 

9358  ! 

7 

9419 

9481 

9542 

9604 

9665 

9726 

9788 

9849 

9911 

9972  1 

8 

850033 

0095 

0156 

0217 

0279 

0340 

0401 

0462 

0524 

0585  i 

9 

0046 

0707 

0769 

0830 

0891  | 

0952 

1014 

1075  J 

1136 

1197  , 

710 

1258 

1320 

1381 

1442 

1503  | 

1564 

1625 

1686 

1747 

1809  ! 

1 

1870 

1931 

1992 

2053 

2114 

2175 

2236 

2297  1 

2358 

2419  1 

61 

2 

2480 

2541 

2602 

2663 

2724 

2785 

2846 

2907  i 

2968 

3029  ! 

3 

3090. 

3150 

3211 

3272 

3333 

3394 

3455 

3516  j 

3577 

3637  1 

4 

3698' 

3759 

3820 

3881 

3941 

4002 

4063 

4124 

4185 

4245 

5 

4306 

4367 

4428 

4488 

4549 

4610 

4670 

4731 

4792 

4852  ! 

6 

4913 

4974 

5034 

5095 

5156 

5216 

5277 

5337 

5398 

5459  ! 

7 

5519 

5580 

5640 

5701 

5761 

5822 

5882 

5943 

6003 

6064  ! 

8 

6124 

6185 

6245 

6306 

6366 

6427 

6487 

6548  1 

6608 

6668  I 

9 

6729 

6789 

1 

6850 

6910 

6970 

7031 

7091 

7152 

7212 

7272 

Proportional  Parts. 


Diff. 

1 

2 

3 

4 

5 

6 

7 

8 

9 

65 

6.5 

13.0 

19.5 

26.0 

32.5 

39.0 

45.5 

52.0 

58.5 

64 

6.4 

12.8 

19.2 

25.6 

1 32.0 

38.4 

44.8 

51.2 

57.6 

63 

6.3 

12.6 

18  9 

25.2 

I 31.5 

37.8 

44.1 

50.4 

56.7 

62 

6.2 

12.4 

18.6 

24.8  1 

31.0 

37.2 

43.4 

49.6 

55.8 

61 

6.1 

12.2 

18.3 

24.4 

30.5 

36.6 

42.7 

48.8 

54  9 

60 

6.0 

12.0 

18.0 

24.0  1 

30.0 

36.0 

42.0 

48.0 

54.0 

190 


TABLE  XT.  LOGARITHMS  OF  NUMBERS. 


No. 

720  L.  857.] 

[No.  764  L.  883. 

N. 

0 

1 

2 

a 

4 

5 

6 

7 

8 

9 

Diff. 

720 

857332 

7393 

7453 

7513 

7574 

7634 

7694 

7755 

7815 

7875 

1 

7935 

7995 

8056 

8116 

8176 

8236 

8297 

8357 

8417 

8477 

2 

8537 

8597 

8657 

8718 

8778 

! 8838 

8898 

8958 

9018 

9078 

3 

9138 

9198 

9258 

9318 

9379 

j 9439 

9499 

9559 

9619 

9679 

60 

4 

9739 

9799 

9859 

9918 

9978 

0038 

0098 

0158 

0218 

0278 

5 

860338 

0398 

0458 

0518 

0578 

0637 

0697 

0757 

0817 

0877 

6 

0937 

0996 

1056 

1116 

1176 

1236 

1295 

1355 

1415 

1475 

7 

1534 

1594 

1654 

1714 

1773 

1833 

1893 

1952 

2012 

2072 

8 

2131 

2191 

2251 

2310 

2370 

2430 

2489 

2549 

2608 

2668 

9 

2728 

2?87 

2847 

2906 

2966 

3025 

3085 

3114 

3204 

3263 

730 

3323 

3382 

3442 

3501 

3561 

3620 

3680 

3739 

3799 

3858 

1 

3917 

3977 

4036 

4096 

4155 

4214 

4274 

4333 

4392 

4452 

2 

4511 

4570 

4630 

4689 

4748 

4808 

4867 

4926 

4985 

5045 

3 

5104 

5163 

5222 

5282 

5341 

5400 

5459 

5519 

5578 

5637 

4 

5696 

5755 

5814 

5874 

5933 

5992 

6051 

6110 

6169 

6228 

5 

6287 

6346 

6405 

6465 

6524 

6583 

6642 

6701 

6760 

6819 

6 

6878 

6937 

6996 

7055 

7114 

7173 

7232 

7291 

7350 

7409 

59 

7 

7467 

7526 

7585 

7644 

7703 

7762 

7821 

7880 

7939 

7998 

8 

8056 

8115 

8174 

8233 

8292 

8350 

8409 

8468 

8527 

8586 

9 

8644 

8703 

8762 

8821 

8879 

8938 

8997 

9056 

9114 

91,3 

740 

9232 

9290 

9349 

9408 

9466 

9525 

9584 

9642 

9701 

9760 

1 

9818 

9877 

9935 

9994 

0053 

0111 

0170 

0228 

0287 

0345 

2 

870404 

0462 

0521 

0579 

0638 

0696 

0755 

0813 

0872 

0930 

3 

0989 

1047 

1106 

1164 

1223 

1281 

1339 

1398 

1456 

1515 

4 

1573 

1631 

1690 

1748 

1806 

1865 

1923 

1981 

2040 

2008 

5 

2156 

2215 

2273 

2331 

2389 

2448 

2506 

2564 

2622 

2681 

6 

2739 

2797 

2855 

2913 

2972 

3030 

3088 

3146 

3204 

3262 

7 

3321 

3379 

3437 

3495 

3553 

3611 

3669 

3727 

3785 

3844 

8 

3902 

3960 

4018 

4076 

4134 

4192 

4250 

4308 

4366 

4424 

58 

9 

4482 

4540 

4598 

4656 

4714 

477'2 

4830 

4888 

4945 

5003 

750 

5061 

5119 

5177 

5235 

5293 

1 5351 

5409 

5466 

5524 

5582 

1 

5640 

5698 

5756 

5813 

5871 

5929 

5987 

6045 

6102 

6160 

2 

6218 

6276 

6333 

6391 

6449 

6507 

6564 

6622 

6680 

6737 

3 

6795 

6853 

6910 

6968 

7026 

7083 

7141 

7199 

7256 

7314 

4 

7371 

7429 

7487 

7544 

7602 

7659 

7717 

7774 

7832 

7889 

5 

7947 

8004 

8062 

8119 

8177 

8234 

8292 

8349 

8407 

8464 

6 

8522 

8579 

8637 

8694 

8752 

8809 

8866 

8924 

8981 

9039 

7 

9096 

9153 

9211 

9268 

9325 

9383 

9440 

9497 

9555 

9612 

8 

9669 

9726 

9784 

9841 

9898 

9956 

0013 

0070 

0127 

0185 

9 

880242 

0299 

0356 

0413 

0471 

0528 

0585 

0642 

0699 

0756 

760 

0814 

0871 

0928 

0985 

1042 

1099 

1156 

1213 

1271 

1328 

1 

1385 

1442 

1499 

1556 

1613 

1670 

1727 

1784 

1841 

1898 

57 

2 

1955 

2012 

2069 

2126 

2183 

2240 

2297 

2354 

2411 

2468 

3 

2525 

2581 

2638 

2695 

2752 

2809 

2866 

2923 

2980 

3037 

4 

3093 

3150 

3207 

3264 

3321 

3377 

3434 

3491 

3548 

3605 

Proportional  Parts. 


Diff. 

i 

2 

3 

4 

5 

6 

7 

8 

9 

59 

5.9 

11.8 

17.7 

23.6 

29.5 

35.4 

41.3 

47.2 

53.1 

58 

5.8 

11.6 

17.4 

23  2 

29.0 

34.8 

40.6 

46.4 

52  2 

57 

5.7 

11.4 

17.1 

22*8 

28.5 

34.2 

39.9 

45.6 

51.3 

56 

5.6 

11.2 

16.8 

22.4 

28.0 

33.6 

39.2 

44.8 

50.4 

TABLE  XT.  LOGARITHMS  OF  NUMBERS. 


101 


No.  765  L.  883.]  [No.  809  L.  908. 


N. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

765 

883661 

3718 

3775 

3832 

3888 

3945 

4002 

4059 

4115 

4172 

6 

4229 

4285 

4342 

4399 

4455 

4512 

4569 

4625 

4682 

4739 

7 

4795 

4852 

4909 

4965 

5022 

5078 

5135 

5192 

5248 

5305 

8 

5361 

5418 

5474 

5531 

5587 

5644 

5700 

5757 

5813 

5870 

9 

5926 

5983 

6039 

6096 

6152 

6209 

6265 

6321 

6378 

6434 

770 

6491 

6547 

6604 

6660 

6716 

6773 

6829 

6885 

6942 

6998 

1 

7054 

7111 

7167 

7223 

7280 

7336 

7392 

7449 

7505 

7561 

2 

7617 

7674 

7730 

7786 

7842 

7898 

7955 

8011 

8067 

8123 

3 

8179 

8236 

8292 

8348 

8404 

8460 

8516 

8573 

8629 

8685 

4 

8741 

8797 

8853 

8909 

8965 

9021 

9077 

9134 

9190 

9246 

5 

9302 

9358 

9414 

9470 

9526 

9582 

9638 

9694 

9750 

9806 

6 

9862 

9918 

9974 

0030 

0086 

0141 

0197 

0253 

0309 

0365 

7 

890421 

0477 

0533 

0589 

0645 

0700 

0756 

0812 

0868 

0924 

8 

0980 

1035 

1091 

1147 

1203 

1259 

1314 

1370 

1426 

1482 

9 

1537 

1593 

1649 

1705 

1760 

1816 

187'2 

1928 

1983 

2039 

780 

2095 

2150 

2206 

2262 

2317 

2373 

2429 

2484 

2540 

2595 

1 

2651 

2707 

2762 

2818 

2873 

2929 

2985 

3040 

3096 

3151 

2 

3207 

3262 

3318 

3373 

3429 

3484 

3540 

3595 

3651 

3706 

3 

3762 

3817 

3873 

3928 

3984 

4039 

4094 

4150 

4205 

4261 

4 

4316 

4371 

4427 

4482 

4538 

4593 

4648 

4704 

4759 

4814 

5 

4870 

4925 

4980 

5036 

5091 

5146 

5201 

5257 

5312 

5367 

6 

5423 

5478 

5533 

5588 

5644 

5699 

5754 

5809 

5864 

5920 

7 

5975 

6030 

6085 

6140 

6195 

6251 

6306 

6361 

6416 

6471 

8 

6526 

6581 

6636 

6692 

6747 

6802 

6857 

6912 

6967 

7022 

9 

7077 

7132 

7187 

7242 

7297 

7352 

7407 

7462 

7517 

7572 

790 

7627 

7682 

7737 

7792 

7847 

7902 

7957 

8012 

8067 

8122 

1 

8176 

8231 

8286 

8341 

8396 

8451 

8506 

8561 

8615 

8670 

2 

8725 

8780 

8835 

8890 

8944 

8999 

9054 

9109 

9164 

9218 

3 

9273 

9328 

9383 

9437 

9492 

9547 

9602 

9656 

9711 

9766 

4 

9821 

9875 

9930 

9985 

0039 

0094 

0149 

0203 

0258 

0312 

5 

900367 

0422 

0476 

0531 

0586 

0640 

0695 

0749 

0804 

0859 

6 

0913 

0968 

1022 

1077 

1131 

1186 

1240 

1295 

1349 

1404 

7 

1458 

1513 

1567 

1622 

1676 

1731 

1785 

1840 

1894 

1948 

8 

2003 

2057 

2112 

2166 

2221 

2275 

2329 

2384 

2438 

2492 

9 

2547 

2601 

2655 

2710 

2764 

2818 

2873 

2927 

2981 

3036 

800 

3090 

3144 

3199 

3253 

3307 

3361 

3416 

3470 

3524 

3578 

1 

3633 

3687 

3741 

37’95 

3849 

3904 

3958 

4012 

4066 

4120 

2 

4174 

4229 

4283 

4337 

4391 

4445 

4499 

4553 

4607 

4661 

3 

4716 

4770 

4824 

4878 

4932 

4986 

5040 

5094 

5148 

5202 

4 

5256 

5310 

5364 

5418 

5472 

5526 

5580 

5634 

5688 

5742 

5 

5796, 

5850 

5904 

5958 

6012 

6066 

6119 

6173 

6227 

6281 

6 

6335 

6389 

6443 

6497 

6551 

6604 

6658 

6712 

6766 

6820 

7 

6874 

6927 

6981 

7035 

7089 

7143 

7196 

7250 

7304 

7358 

8 

7411 

7465 

7519 

7573 

7626 

7680 

7734 

7787 

7841 

7895 

9 

7949 

8002 

8056 

8110 

8163 

8217 

8270 

8324 

8378 

8431 

Proportional  Parts. 


Diff. 

1 

2 

3 

4 

5 

6 

7 

8 

9 

57 

5.7 

11.4 

17.1 

22.8 

28.5 

34.2 

39.9 

45.6 

51.3 

56 

5.6 

11.2 

16.8 

22.4 

28.0 

33.6 

39.2 

44.8 

50.4 

. 55 

5.5 

11.0 

i 16.5 

22.0 

27.5 

33.0 

38.5 

44.0 

49.5 

| 54 

5.4 

10.8 

1 16.2 

21.6 

27.0 

32.4 

37.8 

43.2 

48.6 

192 


TABLE  XI.  LOGARITHMS  OF  NUMBERS, 


No.  810  L.  908.]  |No.  854  L.  931. 


N. 

0 

1 

1 

2 

3 

4 || 

1 

5 

6 

7 

8 ! 

9 

Diff. 

810 

908485 

8539 

8592 

8646 

8699  !! 

8753 

8807  | 

8860 

8914 

8967 

1 

9021 

9074 

9128 

9181 

9235  || 

9289 

9342 

9396 

9449 

9503 

2 

9556 

9610 

9663 

9716 

9770  ! 

9823 

9877 

9930 

9984 

0037 

3 

910091 

0144 

0197 

0251 

0304 

0358 

0411 

0464 

0518 

0571 

4 

0624 

0678 

0731 

0784 

0838  1 

0891 

0944 

0998 

1051 

1104 

5 

1158 

1211 

1264 

1317 

1371 

1424 

1477 

1530 

1584 

1637 

6 

1690 

1743 

1797 

1850 

1903 

1956 

2009 

2063 

2116 

2169 

7 

2222 

2275 

2328 

2381 

2435  ! 

2488 

2541 

2594 

2647 

2700 

8 

2753 

2806 

2859 

2913 

2966 

| 3j19 

3072 

3125 

3178 

3231 

9 

3284 

3337 

3390 

3443 

3496 

! 3549 

3602 

3655 

3708 

3761 

53 

820 

3814 

3867 

3920 

3973 

4026 

4079 

4132 

4184 

4237 

4290 

1 

4343 

4396 

4449 

4502 

4555 

1 4608 

4G60 

4713 

4766 

4819 

2 

4872 

4925 

4977 

5030 

5083 

5136 

5189 

5241  | 

5294 

5347 

3 

5400 

5453 

5505 

5558 

5611 

5664 

5716 

5769  | 

5822 

5875 

4 

5927 

5980 

6033 

6085 

6138 

6191 

6243 

6296 

6349 

6401 

5 

6454 

6507 

6559 

6612 

6664 

0717 

6770 

(822 

6875 

6927 

6 

6980 

7033 

7085 

7138 

7190 

7243 

7295 

7348 

7400 

7453 

7 

7506 

7558 

7611 

7663 

7716 

7768 

7820 

7873 

7925 

7978 

8 

8030 

8083 

8135 

8188 

8240  | 

8293 

8345 

8397 

8450 

8502 

9 

8555 

8607 

8659 

8712 

8764 

8816 

8869 

8921 

8973 

9026 

830 

9078 

9130 

9183 

9235 

9287 

; 9340 

9392 

9444 

9496 

9549 

1 

9601 

9653 

9706 

9758 

9810 

9862 

9914 

9967 

0019 

0071 

2 

920123 

0176 

0228 

0280 

0332 

0384 

0436 

0489 

0541 

0593 

3 

0645 

0697 

0749 

0801 

0853 

0906 

0958 

1010 

1062 

1114 

52 

4 

1166 

1218 

1270 

1322 

1374 

1426 

1478 

1530 

1582 

1634 

5 

1686 

1738 

1790 

1842 

1894 

1946 

1998 

2050 

2102 

2154 

6 

2206 

2258 

2310 

2362 

2414 

2466 

2518 

2570 

2622 

2674 

7 

2725 

2777 

2829 

2881 

2933 

•2985 

3037 

3089 

3140 

3192 

8 

3244 

3296 

3348 

3399 

3451 

3503 

3555 

3607 

3658 

3710 

9 

3762 

3814 

3865 

3917 

3969 

4021 

4072 

4124 

4176 

4228 

840 

4279 

4331 

4383 

4434 

4486 

4538 

4589 

4641 

4693 

4744 

1 

4796 

4848 

4899 

4951 

5003 

5054 

5106 

5157 

5209 

5261 

2 

5312 

5364 

5415 

5467 

5518 

5570 

5621 

5673 

5725 

5776 

3 

5828 

5879 

5931 

5982 

6034 

6085 

6137 

6188 

6240 

6291 

4 

6342 

6394 

6445 

6497 

6548 

6600 

6651 

6702 

6754 

6805 

5 

6857 

6908 

6959 

7011 

7062 

7114 

7165 

7216 

7268 

7319 

6 

7370 

7422 

7473 

7524 

7576 

7627 

7678 

7730 

7781 

7832 

7 

7883 

7935 

7986 

8037 

8088 

8140 

8191 

8242 

8293 

8345 

8 

8396 

8447 

8498 

8549 

8601 

8652 

8703 

8754 

8805 

8857 

9 

8908 

8959 

9010 

9061 

9112 

9163 

9215 

9266 

| 9317 

9368 

850 

9419 

9470 

9521 

9572 

9623 

! 9674 

9725 

9776 

9827 

9879 

51 

noon 

QQQi 

I 

yUoU 

yyoi 

0032 

0083 

0134 

0185 

0236 

0287 

I 0338 

0389 

2 

930440 

0491 

0542 

0592 

0643 

0694 

0745 

0796 

1 0847 

0898 

3 

0949 

1000 

1051 

1102 

1153 

1204 

1254 

1305 

1356 

1407 

4 

1458 

1509 

1560 

1610 

1661 

1712 

1 

1763 

1814 

1865 

1915 

Proportional  Parts. 


Diff. 

1 

2 

3 

4 

5 

6 

7 

8 

9 

53 

5.3 

10.6 

15.9 

21.2 

26.5 

31.8  • 

37.1 

42.4 

47.7 

52 

5.2 

10.4 

15.6 

20.8 

26.0 

31.2 

36.4 

41.6 

46.8 

51 

5.1 

10.2 

15.3 

20.4 

25.5 

30.6 

35.7 

40.8 

45.9 

50 

5.0 

10.0 

15.0 

20.0 

25.0 

30.0 

35.0 

40.0 

45.0 

TABLE  XT.  LOGARITHMS  OF  NUMBERS. 


193 


Mo.  855  L.  881.1  [No.  899  L.  954. 


N. 

0 

1 

2 

3 

4 

8 

6 

7 

8 

9 

Diff. 

355 

931966 

2017 

2068 

2118 

2169 

2220 

2271 

2322 

2372 

2423 

6 

2474 

2524 

2575 

2626 

2677 

2727 

2778 

2829 

2879 

2930 

2981 

3031 

3082 

3133 

3183 

3234 

3285 

3335 

3386 

3437 

8 

3487 

3538 

3589 

3639 

3690 

3740 

3791 

3841 

3892 

3943 

9 

3993 

4044 

4094 

4145 

4195 

4246 

4296 

4347 

4397 

4448 

8C0 

4498 

4549 

4599 

4650 

4700 

4751 

4801 

4852 

4902 

4953 

1 

5003 

5054 

5104 

5154 

5205 

5255 

5306 

5356 

5406 

5457 

2 

5507 

5558 

5608 

5658 

5709 

5759 

5809 

5860 

5910 

5960 

3 

6011 

6061 

6111 

6162 

6212 

6262 

6313 

6363 

6413 

6463 

4 

6514 

6564 

6614 

6665 

6715 

6765 

6815 

6865 

6916 

6966 

5 

7016 

7066 

7116 

7167 

7217 

7267 

7317 

7367 

7418 

7468 

6 

7518 

7568 

7618 

7668 

7718 

7769 

7819 

7869 

7919 

7969 

50 

7 

8019 

8069 

8119 

8169 

8219 

8269 

8320 

8370 

8420 

8470 

8 

8520 

8570 

8620 

8670 

8720 

8770 

8820 

8870 

8920 

8970 

9 

9020 

9070 

9120 

9170 

9220. 

9270 

9320 

9369 

9419 

9469 

870 

9519 

9569 

9619 

9669 

9719 

9769 

9819 

9869 

9918 

9968 

1 

940018 

0068 

0118 

0168 

0218 

0267 

0317 

0367 

0417 

0467 

2 

0516 

0566 

0616 

0666 

0716 

0765 

0815 

0865 

0915 

0964 

3 

1014 

1064 

1114 

1163 

1213 

1263 

1313 

1362 

1412 

1462 

4 

1511 

1561 

1611 

1660 

1710 

1760 

1809 

1859 

1909 

1958 

5 

2008 

2058 

2107 

2157 

2207 

2256 

2306 

2355 

2405 

2455 

6 

2504 

2o54 

2603 

2653 

2702 

2752 

2801 

2851 

2901 

2950 

7 

3000 

3049 

3099 

3148 

3198 

3247 

3297 

3346 

3396 

3445 

8 

3495 

3544 

3593 

3643 

3692 

3742 

3791 

3841 

3890 

3939 

9 

3989 

4038 

4088 

4137 

4186 

4236 

4285 

4335 

4384 

4433 

880 

4483 

4532 

4581 

4631 

4680 

4729' 

4779 

4828 

4877 

4927 

1 

4976 

5025 

5074 

5124 

5173 

5222 

5272 

5321 

5370 

5419 

2 

5469 

5518 

5567 

5616 

5665 

5715 

5764 

5813 

5862 

5912 

3 

5961 

6010 

6059 

6108 

6157 

6207 

6256 

6305 

6354 

6403 

4 

6452 

6501 

6551 

6600 

6649 

6698 

6747 

6796 

6845 

6894 

5 

6943 

6992 

7041 

7090 

7139 

7189 

7238 

7287 

7336 

7385 

49 

6 

7434 

7483 

7532 

7581 

7630 

7679 

7728 

7777 

7826 

7875 

7 

7924 

7973 

8022 

8070 

8119 

8168 

8217 

8266 

8315 

8364 

8 

8413 

8462 

8511 

8560 

8608 

8657 

8706 

8755 

8804 

8853 

9 

8902 

8951 

8999 

9048 

9097 

9146 

9195 

9244 

9292 

9341 

390 

1 

9390 

9878 

9439 

9926 

9488 

9975 

9536 

9585 

9634 

9683 

9731 

9780 

9829 

0024 

0511 

0073 

0560 

0121 

0608 

0170 

0219 

0267 

0316 

2 

950365 

0414 

0462 

0657 

0706 

0754 

0803 

3 

0851 

0900 

0949 

0997 

1046 

1095 

1143 

1192 

1240 

1289 

4 

1338 

1386 

1435 

1483 

1532 

1580 

1629 

1677 

1726 

1775 

5 

1823 

1872 

1920 

1969 

2017 

2066 

2114 

2163 

2211 

2260 

6 

2303 

2356 

2405 

2453 

2502 

2550 

2599 

2647 

2696 

2744 

7 

2792 

2841 

2889 

2938 

2986 

3034 

3083 

3131 

3180 

3228 

8 

3276 

3325 

3373 

3421 

3470 

3518 

3566 

3615 

3663 

3711 

9 

3760 

3808 

3856 

3905 

3953 

4001 

4049 

4098 

4146 

4194 

Proportional  Parts. 


Diff. 

1 

2 

3 

4 

5 

6 

7 

8 

9 

1 

51 

5.1 

10.2 

15.3 

20.4 

25.5 

30.6 

35.7 

40.8 

45.9 

50 

5.0 

10.0 

15.0 

20.0 

25.0 

30.0 

35.0 

40.0 

45.0 

49 

4.9 

9.8 

14.7 

19.6 

24.5 

29.4 

34.3 

39.2 

44.1 

Lii_ 

4.8 

9.6 

14.4 

19.2 

24.0 

28.8 

33.6 

38.4 

43.2 

194 


TABLE  XI.  LOGARITHMS  OF  NUMBERS, 


No  900  L.  954.]  [No.  944  L.  975. 


N. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

Diff. 

900 

954243 

4291 

4339 

4387 

4435 

4484 

4532 

4580 

4628 

4677 

1 

4725 

4773 

4821 

4869 

4918 

4966 

5014 

5062 

5110 

5158 

2 

5207 

5255 

5303 

5351 

5399 

5447 

5495 

5543 

5592 

5640 

3 

5688 

5736 

5784 

5832 

5880 

5928 

5976 

6024 

6072 

6120 

4 

6168 

6216 

6265 

6313 

6361 

6409 

6457 

6505 

6553 

6601 

48 

5 

6649 

6697 

6745 

6793 

6840 

6888 

6936 

6984 

7032 

7080 

6 

7128 

7176 

7224 

7272 

7320 

7368 

7416 

7464 

7512 

7559 

7 

7607 

7655 

7703 

7751 

7799 

7847 

7894 

7942 

7990 

8038 

8 

8086 

8134 

8181 

8229 

8277 

8325 

8373 

8421 

8468 

8516 

9 

8564 

8612 

8659 

8707 

8755 

8803 

8850 

8898 

8946 

8994 

910 

9041 

9089 

9137 

9185 

9232 

9280 

9328 

9375 

9423 

9471 

1 

2 

9518 

9995 

9566 

9614 

9661 

9709 

9757 

9804 

9852 

9900 

9947 

0042 

0090 

0138 

0185 

0233 

0280 

0328 

0876 

0423 

3 

960471 

0518 

0566 

0613 

0661 

. 0709 

0756 

0804 

0851 

0899 

4 

0946 

0994 

1041 

1089 

1136 

1184 

1231 

1279 

1326 

1374 

5 

1421 

1469 

1516 

1563 

1611 

1658 

1706 

1753 

1801 

1848 

6 

1895 

1943 

1990 

2038 

2085 

2132 

2180 

2227 

2275 

2322 

7 

2369 

2417 

2464 

2511 

2559 

2606 

2653 

2701 

2748 

2795 

8 

2843 

2890 

2937 

2985 

3032 

3079 

3126 

3174 

3221 

3268 

9 

3316 

3363 

3410 

3457 

3504 

3552 

3599 

3646 

3693 

3741 

920 

3788 

3835 

3882 

3929 

3977 

4024 

4071 

4118 

4165 

4212 

1 

4260 

4307 

4354 

4401 

4448 

4495 

4542 

4590 

4637 

4684 

2 

4731 

4778 

4825 

4872 

4919 

4966 

5013 

5061 

5108 

5155 

3 

5202 

5249 

5296 

5343 

5390 

5437 

5484 

5531 

5578 

5625 

4 

5672 

5719 

5766 

5813 

5860 

5907 

5954 

6001 

6048 

6095 

47 

5 

6142 

6189 

6236 

6283 

6329 

6376 

6423 

6470 

6517 

6564 

6 

6611 

6658 

6705 

6752 

6799 

6845 

6892 

6939 

6986 

7033 

7 

7080 

7127 

7173 

7220 

7267 

7314 

7361 

7408 

7454 

7501 

8 

7548 

7595 

7642 

7688 

7735 

7782 

7829 

7875 

7922 

7969 

9 

8016 

8062 

8109 

8156 

8203 

8249 

8296 

8343 

8390 

8436 

930 

8483 

8530 

8576 

8623 

8670 

8716 

8763 

8810 

8856 

8903 

1 

8950 

8996 

9043 

9090 

9136 

9183 

9229 

9276 

9323 

9369 

2 

3 

9416 

9882 

9463 

9928 

9509 

9975 

9556 

9602 

9649 

9695 

9742 

9789 

9835 

0021 

0068 

0114 

0161 

0207 

0254 

0300 

4 

970347 

0393 

0440 

0486 

0533 

0579 

0626 

0672 

0719 

0765 

5 

0812 

0858 

0904 

0951 

0997 

1044 

1090 

1137 

1183 

1229 

6 

1276 

1322 

1369 

1415 

1461 

1508 

1554 

1601 

1647 

1693 

7 

1740 

1786 

1832 

1879 

1925 

1971 

2018 

2064 

2110 

2157 

8 

2203 

2249 

2295 

2342 

2388 

2434 

2481 

2527 

2573 

2619 

9 

2666 

2712 

2758 

2804 

2851 

2897 

2943 

2989 

3035 

3082 

940 

3128 

3174 

3220 

3266 

3313 

3359 

3405 

3451 

3497 

3543 

1 

3590 

3636 

3682 

3728 

3774 

3820 

'3866 

3913 

3959 

4005 

2 

4051 

4097 

4143 

4189 

4235 

4281 

4327 

4374 

4420 

4466 

3 

4512 

4558 

4604 

4650 

4696 

4742 

4788 

4834 

4880 

4926 

4 

4972 

5018 

5064 

5110 

5156 

5202 

5248 

5294 

5340 

5386 

46 

Proportional  Parts. 


Diff. 

1 

2 

3 

4 

5 

6 

7 

8 

9 

47 

4.7 

9.4 

14.1 

18.8 

23.5 

28.2 

32.9 

37.6 

42.3 

46 

4.6 

9.2 

13.8 

18.4 

23.0 

27.6 

32.2 

36.8 

41.4 

TABLE  XI.  LOGARITHMS  OF  NUMBERS.  195 

No.  945  L.  975.]  [No.  989  L.  995 


N. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

Diff. 

945 

975432 

5478 

5524 

5570 

5616 

5662 

5707 

5753 

5799 

5845 

6 

5891 

£937 

5983 

6029 

6075 

6121 

6167 

6212 

6258 

6304 

7 

6350 

6396 

6442 

6488 

6533 

6579 

6625 

6671 

6717 

6763 

8 

6808 

6854 

6900 

6946 

6992 

7037 

7083 

7129 

7175 

7220 

9 

7266 

7312 

7358 

7403 

7449 

7495 

7541 

7586 

7632 

7678 

950 

7724 

7769 

7815 

7861 

7906 

7952 

7998 

8043 

8089 

8135 

1 

8181 

8226 

8272 

8317 

8363 

8409 

8454 

8500 

8546 

8591 

2 

8637 

8683 

8728 

8774 

8819 

8865 

8911 

8956 

9002 

9047 

3 

9093 

9138 

9184 

9230 

9275 

9321 

9366 

9412 

9457 

9503 

4 

9548 

9594 

9639 

9685 

9730 

9776 

9821 

9867 

9912 

9958 

5 

980003 

0049 

0094 

0140 

0185 

0231 

0276 

0322 

0367 

0412 

6 

6458 

0503 

0549 

0594 

0640 

0685 

0730 

0776 

0821 

0867 

7 

0912 

0957 

1003 

1048 

1093 

1139 

1184 

1229 

1275 

1320 

8 

1366 

1411 

1456 

1501 

1547 

1592 

1637 

1683 

1728 

1773 

9 

1819 

1864 

1909 

1954 

2000 

2045 

2090 

2135 

2181 

2226 

960 

2271 

2316 

2362 

2407 

2452 

2497 

2543 

2588 

2633 

2678 

1 

2723 

2769 

2814 

2859 

2904 

2949 

2994 

3040 

3085 

3130 

2 

3175 

3220 

3265 

3310 

3356 

3401 

3446 

3491 

3536 

3581 

3 

3626 

3671 

3716 

3762 

3807 

3852 

3897 

3942 

3987 

4032 

4 

4077 

4122 

4167 

4212 

4257 

4302 

4347 

4392 

4437 

4482 

5 

4527 

4572 

4617 

4662 

4707 

4752 

4797 

4842 

4887 

4932 

45 

6 

4977 

5022 

5067 

5112 

5157 

5202 

5247 

5292 

5337 

5382 

7 

5426 

5471 

5516 

5561 

5606 

5651 

5696 

5741 

5786 

5830 

8 

5875 

5920 

5965 

6010 

6055 

6100 

6144 

6189 

6234 

6279 

9 

6324 

6369 

6413 

6458 

6503 

6548 

6593 

6637 

6682 

6727 

970 

6772 

68i7 

6861 

6906 

6951 

6996 

7040 

7085 

7130 

7175 

1 

7219 

7264 

7309 

7353 

7398 

7443 

7488 

7532 

7577 

7622 

2 

7666 

7711 

7756 

7800 

7845 

7890 

7934 

7979 

8024 

8068 

3 

8113 

8157 

8202 

8247 

8291 

8336 

8381 

8425 

8470 

8514 

4 

8559 

8604 

8648 

8693 

8737 

8782 

8826 

8871 

8916 

8960 

5 

9005 

9049 

9094 

9138 

9183 

9227 

9272 

9316 

9361 

9405 

6 

9450 

9494 

9539 

9583 

9628 

9672 

9717 

9761 

9806 

9850 

7 

9895 

9939 

9983 

0028 

0072 

0117 

0161 

0206 

0250 

0294 

8 

990339 

0383 

0428 

0472 

0516 

0561 

0605 

0650 

0694 

0738 

9 

0783 

0827 

0871 

0916 

0960 

1004 

1049 

1093 

1137 

1182 

980 

1226 

1270 

1315 

1359 

1403 

1448 

1492 

1536 

1580 

1625 

1 

1669 

1713 

1758 

1802 

1846 

1890 

1935 

1979 

2023 

2067 

2 

2111 

2156 

2200 

2244 

2288 

2333 

2377 

2421 

2465 

2509 

3 

2554 

2598 

2642 

2686 

2730 

2774 

2819 

2863 

2907  • 

2951 

4 

2995 

3039 

3083' 

3127 

3172 

3216 

3260 

3304 

3348 

3392 

5 

3436 

3480 

3524 

3568 

3613 

3657 

3701 

3745 

3789 

3833 

6 

3877 

- 3921 

3965 

4009 

4053 

4097 

4141 

4185 

4229 

4273 

7 

4317 

4361 

4405 

4449 

4493 

4537 

4581 

4625 

4669 

4713 

44 

8 

4757 

4801 

4845 

4889 

4933 

4977 

5021 

5065 

5108 

5152 

9 

5196 

5240 

5284 

5328 

5372 

5416 

5460 

5504 

5547 

5591 

Proportional  Parts. 


Diff. 

1 

2 

3 

4 

5 

6 

7 

8 

9 

46 

4.6 

9.2 

13.8 

18.4 

23.0 

27.6 

32.2 

36.8 

41.4 

45 

4.5 

9.0 

13.5 

18.0 

22.5 

27.0 

. 31.5 

36.0 

40.5 

44 

4.4 

8.8 

13.2 

17.6 

22.0 

26.4 

30.8 

35.2 

39.6 

43 

4.3 

8.6 

12.9 

17.2 

21.5 

•25.8 

30.1 

34.4 

38.7 

196 


TABLE  XI.  LOGARITHMS  OF  NUMBERS, 


No.  990  L.  995.]  [No.  999  L.  999. 


N. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

Diff. 

990 

996635 

5679 

5723 

5767 

5811 

5854 

5898 

5942 

5986 

6030 

1 

6074 

6117 

6161 

6205 

6249 

6293 

6337 

6380 

6424 

6468 

44 

2 

6512 

6555 

6599 

6643 

6687 

6731 

6774 

6818 

6862 

6906 

3 

6949 

6993 

7037 

7080 

7124 

7168 

7212 

7255 

7299 

7343 

4 

7386 

7430 

7474 

7517 

7561 

7605 

7648 

7692 

7736 

7779 

5 

7823 

7867 

7910 

7954 

7998 

8041 

8085 

8129 

8172 

8216 

6 

8259 

8303 

8347 

8390 

8434 

8477 

8521 

8564 

8608 

8652 

7 

8695 

8739 

8782 

8826 

8869 

8913 

8956 

9000 

9043 

9087 

8 

9131 

9174 

9218 

9261 

9305 

9348 

9392 

9435 

9479 

9522 

9 

9565 

9609 

9652 

9696 

9739 

9783 

9826 

9870 

9913 

9957 

43 

Constant  Numbers  and  their  Logarithms. 


Symbol. 

Number. 

Logarithm. 

7T 

3.141  592  653  590 

0.497  149  872  694 

27T 

6.283  185  307  180 

0.798  179  868  358 

Sn 

9.424  777  960  769 

0.974  271  127  414 

47 r 

12.566  370  614  359 

1.099  209  864  022 

5l T 

15.707  963  267  950 

1.196  119  877  030 

6tt 

18.849  555  921  539 

1 .275  301  123  078 

7 it 

21.991  148  575  119 

1.342  247  912  708 

Sn 

25.132  741  228  718 

1.400  239  859  686 

9n 

28.274  333  882  308 

1.451  3921182  133 

In 

0.523  598  775  598 

T. 71 8 998  622  310 

\ 77 

0.785  398  163  397 

y .895  089  881  366 

1.570  796  326  795 

0.196  119  877  030 

4.187  790  204  786 

0.622  088  609  302 

7T2 

9.869  604  401  089 

0.994  299  745  388 

7 T3 

31.006  276  680  293 

1.491  449  618  082 

V rr 

1.772  453  850  906 

0.248  574  936  347 

V 7T 

1.464  591  887  562 

0.165  716  624  231 

1/7T 

0.318  309  886  184 

T.502  850  127  306 

180/7T 

57.295  779  513  025 

1.758  122  632  409 

1/7T2 

0.101  321  183  642 

y.005  700  254  612 

1/  V 7T 

0.564  189  583  548 

T.751  425  063  653 

lOg^TT 

1.144  729  885  849 

0.058  703  021  240 

arc  1° 

0.017  453  292  520 

IT. 241  877  367  591 

sin  1° 

0.017  452  406  417 

■2.241  855  318  418 

arc  V 

0.000  290  888  209 

y. 463  726  117  207 

sin  1' 

0.000  290  888  205 

y.463  726  111  082 

arc  1" 

0.000  004  848  137 

y.685  574  866  824 

sin  1" 

0.000  004  848  137 

-6\685  574  866  822 

e 

2.718  281  828  459 

0.434  294  481  903 

M 

0.434  294  481  903 

T.637  784  311  301 

1/M 

2.302  585  092  994 

0.362  215  688  699 

i/2 

1.414  213  562  373 

0.150  514  997  832 

V3 

1.732  050  807  569 

0.238  560  627  360 

V5 

2.236  067  977  477 

0.349  485  002  168 

Table  XII, 


LOGARITHMIC  SINES,  COSINES,  TANGENTS, 
AND  COTANGENTS 

TO 


SIX  DECIMAL  PLACES. 


0‘ 


TABLE  XII.  LOGARITHMIC  SINES, 


179 


" 

' 

Sine. 

S T 

Tang. 

Cotang. 

C 

Dl" 

Cosine. 

/ 

0 

60 

120 

180 

240 

300 

360 

420 

480 

540 

600 

660 

720 

780 

840 

900 

960 

1020 

1080 

1140 

1200 

1260 

1320 

1380 

1440 

1500 

1560 

1620 

1680 

1740 

1800 

1860 

1920 

1980 

2040 

2100 

2160 

2220 

2280 

2340 

2400 

2460 

2520 

2580 

2640 

2700 

2760 

2820 

2880 

2940 

3000 

3060 
3120 
3180 
3240 
3300 
3 560 
3120 
3180 
3140 
3600 

0 

1 

2 

3 

4 

5 

6 

8 

9 

10 

11 

12 

13 

14 

15 

16 

17 

18 

19 

20 

21 

22 

23 

24 

25 

26 

27 

28 

29 

30 

31 

32 

33 

34 

35 

36 

37 

38 

39 

40 

41 

42 

43 

44 

45 

46 

47 

48 

49 

50 

51 

52 

53 

54 

55 

56 

57 

58 

59 

60 

Inf.  neg. 

6.463726 

.764756 

6.940847 

7.065786 

.162696 

.241877 

.308824 

.366816 

.417968 

.463726 

7.505118 

.542906 

.577668 

.609853 

.639816 

.667845 

.694173 

.718997 

.742478 

.764754 

7.785943 

.806146 

.825451 

.843934 

.861662 

.878695 

.895085 

.910879 

.926119 

.940842 

7.955082 
. 96887'0 
.982233 

7.995198 

8.007787 

.020021 

.031919 

.043501 

.054781 

.065776 

8.076500 

.086965 

.097183 

.107167 

.116926 

.126471 

.135810 

.144953 

.153907 

.162681 

8.171280 

.179713 

.187985 

.196102 

.204070 

.211895 

.219581 

.227134 

.234557 

8.241855 

4.685 
575  I 575 
575  575 

575  575 

575  575 

575  575 

575  575 

575  575 

575  575 

574  576 
574  576 
574  576 

574  576 
574  577 
574  577 
574  577 
573  578 
573  578 
573  578 
573  579 

573  579 

572  580 

572  580 
572  ! 581 
572  581 
571  582 

571  583 
571  583 
570  584 

570  584 
570  585 

569  586 

569  587 
569  587 
568  588 
568  589 
567  590 
567  591 
566  592 
566  593 
566  593 
565  594 

565  595 
564  596 

564  598 
563  599 
562  600 

562  601 
561  602 

561  603 

5(50  604 

560  605 

559  607 
558  608 
558  (509 

557  I 611 
556  612 
556  (513 

555  ! 615 
554  61(5 

554  618 

553  l!  619 
4.685 

Inf.  neg. 

6.463726 

.764756 

6.940847 

7 f065786 
.162696 
.241878 
.308825 
.366817 
.417970 
.463727 

7.505120 

.542909 

.577672 

.609857 

.639820 

.667849 

.694179 

.719003 

.742484 

.764761 

7.785951 

.806155 

.825460 

.843944 

.861674 

.878708 

.895099 

.910894 

.926134 

.940858 

7.955100 

.968889 

.982253 

7.995219 

8.007809 

.020044 

.031945 

.043527 

.054809 

.065806 

8.076531 

.086997 

.097217 

.107203 

.116963 

.126510 

.135851 

.144996 

.153952 

.162727 

8.171328 
.179763 
.188036 
.196156 
.204126 
.211953 
.219641 
.227195 
. 234621 

8.241921 

Inf.  pos. 

13.536274 

.235244 

13.059153 

12.934214 

.837304 

.758122 

.691175 

.633183 

.582030 

.536273 

12.494880 

.457091 

! 390143 
.360180 
.332151 
.305821 
.280997 
.257516 
.235239 

12.214049 

.193845 

.174540 

.156056 

.138326 

.121292 

.104901 

.089106 

.073866 

.059142 

12.044900 

.031111 

.017747 

12.004781 

11.992191 

.979956 

.968055 

.956473 

.945191 

.934194 

11 .923469 
.913003 
.902783 
.892797 
.883037 
. 873490 
.864149 
.855004 
.846048 
.837273 

11.828672 

.820237 

.811964 

.803844 

.795874 

.788047 

.780359 

.772805 

.765379 

11.758079 

15.314 

425 

425 

425 

425 

425 

425 

425 

425 

424 

424 

424 

424 

423 

423 

423 

422 

422 

422 

421 

421 

420 

420 

419 

419 

418 

417 

417 

416 

416 

415 

414 

413 

413 

412 

411 

410 

409 

408 

407 

407 

406 

405 

404 

402 

401 

400 

399 

398 

397 

396 

395 

393 

392 

391 

389 

388 

387 

385 

384 

382 

381 

15.314] 

.02 

.00 

.00 

.00 

.02 

.00 

.02 

.00 

.02 

.00 

.02 

.00 

.02 

.02 

.00 

.02 

.02 

.02 

.02 

.00 

.02 

.02 

.02 

.02 

.03 

.02 

.02 

.02 

.02 

.03 

.02 

.02 

.03 

.02 

.02 

.03 

.02 

.03 

.03 

.02 

,03 

.03 

.02 

.03 

.03 

.03 

.03 

.03 

.03 

.03 

.03 

.03 

.03 

.03 

.03 

ten 

ten 

ten 

ten 

ten 

ten 

9.999999 

.999999 

.999999 

.999999 

.999998 

9.999998 

.999997 

.999997 

.999996 

.999996 

.999995 

.999995 

.999994 

.999993 

.999993 

9.999992 

.999991 

.999990 

.999989 

.999989 

.999988 

.999987 

.999986 

.999985 

.999983 

9.999982 
.999981 
.999980 
.999979 
. 999977 
.999976 
.999975 
.999973 
.999972 
.999971 

9.999969 

.999968 

.999966 

.999964 

.999963 

.9999(51 

.999959 

.999958 

.999956 

.999954 

9.999952 
.999950 
.999948 
.999946 
. 999944 
.999942 
. 999940 
.999938 
.999986 

9.999934 

60 

59 

58 

57 

56 

55 

54 

53 

52 

51 

50 

49 

48 

47 

46 

45 

44 

43 

42 

41 

40 

39 

38 

37 

36 

35 

34 

33 

32 

31 

30 

29 

28 

27 

26 

25 

24 

23 

22 

21 

20 

19 

18 

17 

16 

15 

14 

13 

12 

11 

10 

9 

8 

6 

5 

4 

3 

2 

1 

0 

// 

' 

Cosine. 

Cotnng. 

Tang. 

Dl" 

Sine. 

' 

COSINES,  TANGENTS,  AND  COTANGENTS.  17g* 


// 

/ 

Sine. 

S 

T 

Tang. 

Cotang. 

C 

Dl" 

Cosine. 

/ | 

4.685 

15.314 

1 

3600  : 

0 

8.241855 

553 

619 

8.241921 

11.758079  | 

381  | 

.03  1 

.05 

.03 

.03 

.05 

.03 

.03 

.05 

.03 

•05 

9.999934 

60 

3660 

1 

.249033 

552 

620 

.249102 

.750898  | 

380 

.999932 

59 

3720 

2 

.256094 

551 

622 

.256165 

.743835 

378  1 

.999929 

58 

3780  | 

3 

.263042 

551 

623 

.263115 

.736885 

377 

.999927 

57 

3840 

4 

.269881 

550 

625 

.269956 

.730044  1 

375 

.999925 

56 

3900 

5 

.276614 

549 

627| 

.276691 

.723309 

373 

.999922 

55 

3960 

6 

.283243 

548 

628 

.283323 

.716677 

372 

.999920 

54 

4020 

7 

.289773 

547 

630 

.289856 

.710144 

370 

.999918 

53 

4080 

8 

.296207 

546 

632 

.296292 

.703708 

368 

.999915 

52 

4140 

9 

.302546 

546 

633 

.302634 

.697366 

367 

.999913 

51 

4200 

10 

.308794 

545 

635 

.308884 

.691116 

365 

.999910 

50 

4260 

11 

8.314954 

544 

637 

8.315046 

11.684954 

363 

.05 
.03 
.05  1 
.05 
.03 
.05  j 
.05 
.05  | 
.05  1 
.05 

9.999907 

49 

4320 

12 

.321027 

543 

638 

.321122 

.678878 

362 

.999905 

48 

4380 

13 

.327016 

542 

640 

.327114 

.672886 

360 

.999902 

47 

4440 

14 

.332924 

541 

642 

.333025 

.666975 

358 

.999899 

46 

4500 

15 

.338753 

540 

644 

.338856 

.661144 

356 

.999897 

45 

4560 

16 

.344504 

539 

646 

.344610 

.655390 

354 

.999894 

44 

4620 

17 

.350181 

539 

648 

.350289 

.649711  i 

352 

.999891 

43 

4680  | 

18 

.355783 

538 

649 

.355895 

.644105 

351 

.999888 

42 

4740  | 

19 

.361315 

537 

651 

.361430 

.638570 

349 

.999885 

41 

4800 

20 

.366777 

536 

653 

.366895 

.633105 

347 

.999882 

40 

4860 

21 

8.372171 

535 

655 

8.372292 

11.627708 

345 

! .05 
j .05 
.05 
.05 
.05 
.05 
.05 
1 .05 
.07 
.05 

9.999873 

39 

4920 

22 

.377499 

534 

657 

.377622 

.622378 

343 

.999876 

38 

4980 

23 

.382762 

533 

659 

.382889 

.617111 

341  ! 

.999873 

37 

5040 

24 

.387962 

532 

661 

.388092 

.611908 

339 

.999870 

36 

5100 

25 

.393101 

531 

663 

.393234 

.606766 

337  1 

.999867 

35 

5160 

26 

.398179 

530 

666 

.398315 

.601685 

334 

.999864 

34 

5220 

27 

.403199 

529 

668 

.403338 

.596662 

332 

.999861 

33 

5280 

28 

.408161 

527 

670 

.408304 

.591696 

330 

.999858 

32 

5340 

29 

.413068 

526 

672 

.413213 

.586787 

328 

.999854 

31 

5400 

30 

.417919 

525  | 

674 

.418068 

.581932 

326 

.999851 

30 

5460 

31 

8.422717 

524  1 

676 

8.422869 

11.577131 

324 

.05 

.07 

.05 

.05 

.07 

.05 

.07 

.05 

.07 

.07 

.05 

.07 

.07 

.07 

.07 

.05 

.07 

.07 

.07 

.07 

9.999848 

29 

5520 

32 

.427462 

523 

679 

.427618 

.572382 

321 

.999844 

28 

5580 

33 

.432156 

522 

681 

.432315 

.567685 

319 

.999841 

27 

5640 

34 

.436800 

521 

683 

.436962 

.563038 

317 

.999838 

26 

5700 

35 

.441394 

520 

685 

.441560 

.558440 

315 

.999834 

25 

5760 

36 

.445941 

518 

688 

.446110 

.553890 

312 

.999831 

24 

5820 

37 

.450440 

517 

690 

.450613 

.549387 

310 

.999827 

23 

5880 

38 

.454893 

516 

693 

.455070 

.544930 

307 

.999824 

22 

5940 

39 

.459301 

515 

695 

.459481 

.540519 

305 

.999820 

21 

6000 

40 

.463665 

514 

697 

.463849 

.536151 

303 

.999816 

20 

6060 

41 

8.467985 

512 

700 

8.468172 

11.531828 

300 

9.999813 

19 

6120 

42 

.472263 

511 

702 

.472454 

.527546 

298 

.999809 

18 

6180 

43 

.476498 

510 

705 

.476693 

.523307 

295 

.999805 

17 

6240 

44 

.480693 

509 

707 

.480892 

.519108 

293 

.999801 

16 

6300 

45 

.484848 

507 

710 

.485050 

.514950 

290 

.999797 

15 

6360 

46 

.488963 

506 

713 

.489170 

.510830 

287 

.999794 

14 

642a 

47 

.493040 

505 

715 

.493250 

.506750 

285 

.999790 

13 

6480 

48 

.497078 

503 

718 

.497293 

.502707 

282 

.999786 

12 

6540 

49 

.501080 

502 

720 

.501298 

.498702 

280 

.999782 

11 

6600 

50 

.505045 

501 

723 

.505267 

• .494733 

277 

.999778 

ia 

6660 

51 

8.508974 

499 

726 

8.509200 

11.490800 

274 

.07 

08 

9.999774 

9 

6720 

52 

.512867 

498 

729 

.513098 

.486902 

271 

!07 

.07 

.07 

.07 

.08 

.07 

.999769 

8 

6780 

53 

.516726 

497 

731 

.516961 

.483039 

269 

.999765 

7 

6840 

54 

.520551 

495 

734 

.520790 

.479210 

266 

.999761 

6 

6900 

55 

. 524343 

494 

737 

524586 

.475414 

263 

.999757 

5 

6960 

56 

.528102 

492 

740 

.528349 

.471651 

260 

.999753 

4 

7020 

57 

.531828 

491 

743 

.532080 

.467920 

257 

.999748 

3 

7080 

58 

.535523 

490 

745 

.535779 

.464221 

255 

.999744 

2 

7140 

59 

.539186 

488 

748 

.539447 

.460553 

252 

J ‘.08 

1 

.999740 

1 

7200 

1 60 

8.542819 

487 

| 4.i 

?51 

685 

| 8.543084 

11.456916 

1 

249 

[15.314 

9.999735 

0 

// 

' 

Cosine. 

1“ 

! Cotang. 

1 Tang. 

1 1 

Dl* 

Sine. 

1 ' 

^1° 1 QQ 


88 


2 


TABLE  XII.  LOGARITHMIC  SIXES. 


277' 


/ 

Sine. 

D.  1". 

Cosine. 

D.  1". 

Tang. 

| 

D.  1". 

Cotang. 

/ 

0 

8 542819 

60.05 
59  55 
59.07 
58.58 
58.10 
57.65 

57.20 
56.75 
56.30 
55.87 
55.43 

55.02 

54.60 

54.20 
53.78 

53.40 
53.00 
52.62 
52.23 
51.85 
51.48 

51.13 

50.77 

50.42 

50.05 
49.72 
49.38 

49.05 
48.70 

48.40 

48.05 

Aty 

9.999735 

.07 

.08 

.07 

.08 

.07 

.08 

.07 

.08 

.08 

.08 

.07 

.08 

8.543084 

60.12 

59.62 

59.15 
58.65 
58.20 

57.72 

57.27 
56.83 
56.38 
55.95 

55.52 

55.10 

54.68 

54.27 
53.87 
53.48 
53.08 
52.70 

52.32 
51.93 
51.58 

51.22 

50.85 
50.50 

50.15 

49.80 

49.47 
49.13 

48.80 

48.48 

48.15 

47.85 

47.52 

47.22 
46.92 

46.62 

46.32 
46.02 

45.73 
45.45 
45.17 

11.456916 

60 

1 

.546422 

.999731 

.546691 

.453309 

59 

2 

.549995 

. 999720 

.550268 

.449732 

.58 

3 

.553539 

.999722 

.553817 

.446183 

57 

4 

.557054 

.999717 

. 557336 

.442664 

56 

5 

.560540 

.999713 

.560828 

.439172 

| 55 

6 

.563999 

.999708 

.564291 

.435709 

54 

7 

.567431 

. 9997*04 

.567727 

.43227*3 

i 53 

8 

.570836 

.999699 

.571137 

.428863 

52 

9 

.574214 

.999694 

.574520 

.425480 

51 

10 

11 

. 577566 
8.580892 

.999689 

9.999685 

.577877 

8.581208 

.422123 

11.418792 

50 

49 

12 

.584193 

.999680 

.584514 

.415486 

48 

13 

.587469 

.999675 

.08 

.08 

.08 

.08 

.08 

.08 

.08 

.08 

.10 

.08 

.08 

.08 

.10 

.08 

.10 

.08 

.10 

.08 

.10 

.08 

.10 

.10 

.08 

.10 

.10 

.10 

.10 

.08 

.587795 

.412205 

47 

14 

.590721 

.99967*0 

.591051 

.408949 

46 

15 

.593948 

.999665 

.594283 

.405717 

45 

10 

.597152 

.999660 

.597492 

.402508 

44 

17 

.000332 

.999655 

.600677 

.399323 

43 

18 

.603489 

.999650 

.603839 

.396161 

42 

19 

.606623 

.999645 

.606978 

.393022 

41 

20 

21 

.609734 

8.612823 

.999640 

9.999635 

.610094 

8.613189 

.389906 

11.386811 

40 

39% 

22 

.615891 

.999629 

. 616262 

.383738 

38 

23 

.618937 

.999624 

.619313 

.380687 

37 

24 

.621962 

.999619 

.622343 

.377657 

36 

25 

.624965 

.999614 

.625352 

.374648 

35 

26 

.627948 

.999608 

.628340 

.371660 

34 

27 

.630911 

.999603 

.631308 

.368692 

33 

28 

.633854 

.999597 

.634256 

.365744 

32 

29 

. 636776 

.999592 

.637184 

.362816 

31 

30 

31 

.639680 

8.642563 

.999586 

9.999581 

.640093 

8.642982 

.359907 

11.357018 

30 

29 

32 

.645428 

47.43 

47.13 

46.82 

46.52- 

46.22 

45.92 

45.63 

45.35 

45.07 

.999575 

.645853 

.354147 

28 

33 

.648274 

.999570 

.6487*04 

.351296 

27  . 

34 

.651102 

.999564 

.651537 

.348463 

26 

35 

.653911 

.999558 

.654352 

.345648 

25 

36 

.656702 

.999553 

.657149 

.342851 

24 

37 

.659475 

.999547 

.659928 

.340072 

23 

38 

.662230 

.999541 

.662689 

.337311 

22 

39 

.664968 

.999535 

.665433 

.334567 

21 

40 

.667689 

.999529 

.668160 

.331840 

20 

41 

8.670393 

44.78 

44.52 

44.23 

43.97 

43.70 

43.45 

43.18 

42.92 

42.67 
42.42  i 

42.17 

41 . 93  i 

41.68  | 

9.999524 

.10 

.10 

.10 

.10 

.12 

.10 

.10 

.10 

.10 

.10 

.12 

.10 

.12 

.10 

.10 

.12 

.10 

19 

8.670870 

44.88 

44.60 

44.35 

44.07 
43.80 

43.53 
43.28 

43.03 

42.77 

42.53 

42.27 

42.03 

41.78 
41.57 
41.30 

41.08 
40.85 
40.63 
40.40 

11.329130 

19 

42 

.673080 

.999518 

.673563 

.326437 

18 

43 

. 675751 

.999512 

.67*6239 

.323761 

17 

44 

.678405 

.999506 

.67*8900 

.321100 

16 

45 

.681043 

.999500 

.681544 

.318456 

15 

46 

.683665 

.999493 

.684172 

.315828 

14 

47 

' .686272 

.999487 

.686784 

.313216 

13 

48 

.688863 

.999481 

.689381 

.310619 

12 

49 

.6914*18 

.999475 

.691963 

.308037 

11 

50 

51 

.693998 

8.696543 

• .999469 
9.999463 

.694529 

8.697081 

.305471 

11.302919 

10 

9 

52 

.699073 

.999456 

.699617 

.300383 

8 

53 

.701589 

.999450 

.702139 

.297861 

7 

54 

.704090 

.999443 

.704646 

.295354 

6 

55 

.706577 

41 . 45  1 

41.20 

40.97 

40.75 

40.52 

.999437 

.707140 

.2928(50 

5 

56 

.709049 

.999431 

.7*09618 

.290382 

4 

57 

.711507 

.999424 

.712083 

.287917 

3 

58 

.713952 

.999418 

.714534 

.286466 

2 

59 

.716383 

.999411 

. J/w 
.12 

.716972 

.283028 

1 

60 

8.718800 

40.28 

9.999404 

8.719396 

11.280804 

0 

’“I" 

Cog’^e. 

D 1". 

Sine.  I 

D.  1". 

Cotang,  i 

i).  r.  1 

Tang.  I 

„ ,| 

.1 

87 » 


92' 


9 inn 


COSIHES,  TANGENTS,  AHD  COTANGENTS.  176° 


/ 

Sine. 

D.  1". 

' I 

Cosine,  j 

D.  1". 

Tang. 

D- 1". 

Cotang,  j 7 

0 

1 

2 

3 

4 

5 
G 

7 

8 
9 

10 

11 

12 

13 

14 

15 

16 

17 

18 

19 

20 

21 

22 

23 

24 

25 

26 

27 

28 

29 

30 

31 

32 

33 

34 

35 

36 

37 

38 

39 

40 

41 

42 

43 

44 

45 

46 

47 

48 
< 49 

50 

51 

52 

53 

54 

55 

56 

57 
5£ 
r£ 

6C 

8.718800 

.721204 

.723595 

.725972 

.728337 

.730688 

.733027 

.735354 

.737667 

.739969 

.742259 

8.744536 

.746802 

.749055 

.751297 

.<753528 

.755747 

.757955 

.760151 

.762337 

.764511 

8.766675 
.768828 
.770970 
.773101 
.775223 
.777333 
. 779434 
.781524 
.783605 
.785675 

8 . 787736 
.789787 
.791828 
.793859 
.795881 
.797894 
.799897 
.801892 
.803876 
.805852 

8.807819 

.809777 

.811726 

.813667 

.815599 

.817522 

.819436 

.821343 

.823240 

.825130 

8.827011 
.828884 
.830749 
.832607 
.834456 
i .836297 

.838130 
i .839956 

) .841774 

) 8.843585 

40.07 

39.85 

39.62 

39.42 
39.18 

38.98 
38.78 

38.55 

38.37 

38.17 
37.95 

37.77 

37.55 

37.37 

37.18 

36.98 
36.80 
36.60 

36.43 
36.23 
36.07 

35.88 

35.70 
35.52 

35.37 

35.17 

35.02 
34.83 
34.68 
34.50 
34.35 

34.18 

34.02 
33.85 

33.70 
33.55 

33.38 
33.25 
33.07 
32.93 

32.78 

32.63 
32.48 
' 32.35 
32.20 
* 32.05 
31.90 

31.78 
31.62 
31.50 
31.35 

31.22 
31.08 
30.97 
30.82 
! 30.68 

30.55 
30.43 
’ 30.30 

; 30.18 

9.999404 

.999398 

.999391 

.999384 

.999378 

.999371 

.999364 

.999357 

.999350 

.999343 

.999336 

9.999329 

.999322 

.999315 

.999308 

.999301 

.999294 

.999287 

.999279 

.999272 

.999265 

9.999257 

.999250 

.999242 

.999235 

.999227 

.999220 

.999212 

.999205 

.999197 

.999189 

9.999181 
.999174 
.999166 
.999158 
.999150 
.999142 
.999134 
.999126 
.999118 
.999110 
9.999102 
. 999094 
.999086 
.999077 
.999069 
.999061 
.999053 
.999044 
.999036 
.999027 

'9.999019 
.999010 
.999002 
.998993 
.998984 
. 998976 
.998967 
.998958 
.998950 
9.998941 

.10 

.12 

.12 

.10 

.12 

.12 

.12 

.12 

.12 

.12 

.12 

.12 

.12 

.12 

.12 

.12 

.12 

.13 

.12 

.12 

.13 

.12 

.13 

.12 

.13 

.12 

.13 

.12 

.13 

.13 

.13 

.12 

.13 

.13 

.13 

.13 

.13 

.13 

.13 

.13 

.13 

.13 

.13 

.15 

.13 

.13 

.13 

.15 

.13  . 

.15 

.13 

.15 

.13 

.15 

.15 

.13 

.15 

.15 

: 

8.719396 

.721806 

.724204 

.726588 

.728959 

.731317 

.733663 

.735996 

.738317 

.740626 

.742922 

8.745207 

.747479 

.749740 

.751989 

.754227 

.756453 

.758668 

.760872 

.763065 

.765246 

8.767417 

.769578 

.771727 

.773866 

.775995 

.778114 

.780222 

.782320 

.784408 

.786486 

8.788554 

.790613 

.792662 

.794701 

.796731 

.798752 

.800763 

.802765 

.804758 

.806742 

8.808717 

.810683 

.812641 

.814589 

.816529 

.818461 

.820384 

.822298 

.824205 

.826103 

8.827992 
.829874 
.831748 
.833613 
.835471 
.837321 
.839163 
. 840998 
. 842825 
8.844641 

40.17 

39.97 

39.73 
39.52 

39.30 

39.10 
38.88 

38.68 

38.48 
38.27 
38.08 

37.87 

37.68 

37.48 

37.30 

37.10 
36.92 

36.73 
36.55 
36.35 
36.18 

36.02 

35.82 
35.65 

35.48 

35.32 
35.13 

34.97 
34.80 

34.63 

34.47 

34.32 
34.15 

33.98 

33.83 
33.68 
33.52 

33.37 

33.22 

33.07 
32.92 

32.77 

32.63 

32.47 

32.33 
32.20 
32.05 
31.90 

31.78 

31.63 

31.48 

31.37 

31.23 

31.08 
30.97 
30.83 
30.70 

I 30.58 
’ 30.45 

| 30.32 

11.280604  1 

.278194  | ! 
.275796  1 i 
.273412  1 
.271041  1 
.268683 
.266337 
.264004 
.261683 
.259374 
.257078 

11.254793 
.252521 
.250260 
.248011  j 
.245773  j 
.243547 
.241332 
.239128 
.236935 
.234754 

11.232583 

.230422 

.228273 

.226134 

.224005 

.221886 

.219778 

.217680 

.215592 

.213514 

11.211446 

.209387 

.207338 

.205299 

.203269 

.201248 

.199237 

.197235 

.195242 

.193258 

11.191283 

.189317 

.187359 

.185411 

.183471 

.181539 

.179616 

.177702 

.175795 

.173897 

11.172008 

.170126 

.168252 

.166387 

.164529 

.162679 

.160837 

.159002 

| .157175 

’ 11.155356 

50 

59 

58 

57 

56 

55 

54 

53 

52 

51 
50 

49  - 

48 

47 

46 

45 

44 

43 

42 

41 

40 

39 

38 

37 

36 

35 

34 

33 

32 

31 

30 

29 

28 

27 

26 

25 

24 

23 

22 

21 

20 

19 

18 

17  1 

| 16  1 
i 15 
14 
! 13 
1 12 
11 
10 

9 

8 

7 

6 

5 
4 
3 
2 

6 

/ 

1 Cosine. 

D 1". 

1 Sine. 

D.  1". 

1 Cotang 

. D.  1". 

Tang.  1 ' 

Q£o 

4*  Table  xii.  logarithmic  sixes. 


1 

/ 

Sine. 

D.  1". 

Cosine 

D.  1". 

Tang. 

D.  1". 

Cotang. 

/ 

c 

1 

3 

4 

5 

6 

7 

8 
9 

10 

11 

! 12 
13 

! 14 
! 15 
16 

17 

18 

19 

20 

21 

22 

23 

24 

25 

26 

27 

28 

29 

30 

31 

32 

33 

34 

35 

36 

37 

38 

39 

40 

41 

42 

43 

44 

45 

46 

47 

48 

49 

50 

51 

52 

53 

54 

55 

56 

57 

58 

59 

60  : 

8.843585 

.845387 

.847183 

.848971 

.850751 

.852525 

.854291 

.856049 

.857801 

.859546 

.861283 

8.863014 

.864738 

.866455 

.868165 

.869868 

.871565 

.873255 

.874938 

.876615 

.878285 

8.879949 

.881607 

.883258 

.884903 

.886542 

.888174 

.889801 

.891421 

.893035 

.894643 

8.896246 

.897842 

.899432 

.901017 

.902596 

.904169 

.905736 

.907297 

.908853 

.910404 

8.911949 

.913488 

.915022 

.916550 

.918073 

.919591 

.921103 

.922610 

.924112 

.925609 

8.927100 

.928587 

.930068 

.931544 

.933015 

.934481 

.935942 

.937398 

.938850 

8.940296 

30.03 

29.93 

29.80 

29.67 

29.57 

29.43 

29.30 

29.20 

29.08 

28.95 
28.85 

28.73 
28.62 
28.50 
28.38 
28.28 
28.17 
28.05 

27.95 
27.83 

27.73 

27.63 

27.52 

27.42 

27.32 
27.20 

27.12 
27.00 
26.90 
26.80 
26.72 

26.60 

26.50 

26.42 

26.32 
26.22 

26.12 
26.02 
25.93 

25.85 
25.75 

25.65 

25.57 

25.47 

25.38 

25.30 

25.20 
25.12 
25.03 
24.95 

24.85  | 

24.78 
24.68  1 
24.60 
24.52  ! 
24.43 
24.35  1 
24.27  ; 

24.20  i 
24.10 

9.998941 

.998932 

.998923 

.998914 

.998905 

.998896 

.998887 

.998878 

.998869 

.998860 

.998851 

9.998841 

.998832 

.998823 

.998813 

.998804 

.998795 

.998785 

.998776 

.998766 

.998757 

9.998747 

.998738 

.998728 

.998718 

.998708 

.998699 

.998689 

.998679 

.998669 

.998659 

9.998649 

.998639 

.998629 

.998619 

.998609 

.998599 

.998589 

.998578 

.998568 

.998558 

9.1998548 

.998537 

.998527 

.998516 

.998506 

.998495 

.998485 

.998474 

.998464 

.998453 

9.998442 

.998431 

.998421 

.998410 

.998399 

.998388 

.998377 

.998366 

.998355 

9.998344 

.15 

.15 

.15 

.15 

.15 

.15 

.15 

.15 

.15 

.15 

.17 

.15 

.15 

.17 

.15 

.15 

.17 

.15 

.17 

.15 

.17 

.15 

.17 

.17 

.17 

.15 

.17 

.17 

.17 

.17* 

.17 

.17 

.17 

.17 

.17 

.17 

.17 

.18 

.17 

.17 

.17 

.18 

.17 

.18 

.17 

.18 

.17 

.13 

.17 

.18 

.18 

.18 

.17 

.18 

.18 

.18 

.18 

.18 

.18 

.18 

8.844644 

.846455 

.848260 

.850057 

.851846 

.853628 

.855403 

.857171 

.858932 

.860686 

.862433 

8.864173 

.865906 

.867632 

.869351 

.871064 

.872770 

.874469 

.876162 

.877849 

.879529 

8.881202 

.882869 

.884530 

.886185 

.887833 

.889476 

.891112 

.892742 

.894366 

.895984 

8.897596 

.899203 

.900803 

.902398 

.903987 

.905570 

.907147 

.908719 

.910285 

.911846 

8.913401 

.914951 

.916495 

.918034 

.919568 

.921096 

.922619. 

.924136 

.925649 

.927156 

8.928658 
.930155 
.931647 
.933134 
.934616 
.936093 
. 937565 
.939032 
.940494 
8.941952 

30.18 
| 30.08 

29.95 
29.82 
29.70 
29.58 
29.47 
29.35 
29.23 

29.12 

29.00 

28.88 

28.77 
28.65 
28.55 
28.43 
28.32 
28.22 

28.12 

28.00 
27.88 

27.78 
27.68 

27.58 

27.47 

27.38 

27.27 
27.17 
27.07 
26.97 
26.87 

26.78 
26.67 

26.58 

26.48 

26.38 

26.28 
26.20 
26.10 
26.02 
25.92 

25.83 

25.73 

25.63 

25.57 

25.47 

25.38 
. 25.28 

25.22 

25.12 

25.03 

24.95 
24.87 
24.78 
24.70 
24.62 
24.53 
24.45 
24.37 
24.30  . 

11.155356 
.15354.': 
.15174C 
. 149943 
. 148154 
.146372 
.144597 
.142829 
.141068 
.139314 
.137567 
11.135827 
.134094 
132368 
.130649 
.128936 
.127230 
.125531 
.123838 
.122151 
. 120471 
11.118798 
. 117131 
.115470 
.113815 
.112167 
.110524 
.108888 
.107258 
.105634 
.104016 
11.102404 
. 100797 
.099197 
.097602 
.096013 
.094430 
.092853 
.091281 
.089715 
.088154 

11.086599 
.085049 
.083505 
.081966 
.080432 
.078904 
.077381  1 
.075864 
.074351  1 
.072844 

11.071342 
.069845 
.068353 
. 066866 
.065384 
.063907 
.062435 
.0609(58 
.069606 

(1.058048 

5 60 
5 59 

> 58 
5 57 
5 56 
: 55 
’ 1 54 

> ! 53 
| 52 

51 

50 

49 

48 

47 

46 

45 

44 

43 

42 

41 

40 

39 
! 38 
37 
36 
35 
34 
33 
32 
31 
30 

! 29 
28 

I 27 
26 
25 
24 
23 
22 
21 
20 

19 

18 

17 

16 

15 

14 

13 

12 

II 
10 

9 

8 

7 

6 

5 

4 

3 

o 

! 

0 

' 1 

Cosine. 

D.  1".  | 

Sine. 

D.  1". 

Cotang.  | 

D~r7r 

Tang.  1 

/ 

85“ 


94< 


202 


5 


COSTNER,  TANGENTS,  AND  COTANGENTS. 


174< 


/ 

Sine.  | 

D.  1\ 

Cosine. 

D.  1\ 

i Tang. 

D.  1".  I 

Cotang. 

/ 

0 

8.940296 

24.03 
23.93 
23.87 
23. 80 
23.70 
23.63 
23.55 
23.48 
23.40 
23.32 
23.25 

9.998344 

.18 

.18 

.18 

.18 

.18 

.20 

.18 

.18 

.20 

.18 

.20 

1 8.941952 

24.20 

24.13 

24.05 

23.98 

23.90 

23.82 

23.73 

23.67 

23.58 

23.52 

23.45 

11.058048 

60 

1 

.941738 

.998333 

! .943404 

.056596 

59 

.943174 

.998322 

.944852 

.055148 

58 

3 

.944606 

.998311 

.946295 

.053705 

57 

4 

.946034 

.998300 

.947734 

.052266 

56 

5 

.947456 

.998289 

.949168 

.050832 

55 

6 

.948874 

.998277 

.950597 

.049403 

54 

7 

.950287 

.998266 

.952021 

.047979 

53 

•8 

.951696 

.998255 

.953441 

.046559 

52 

9 

.953100 

.998243 

.954856 

.045144 

51 

10 

.954499 

.998232 

.956267 

.043733 

50 

11 

8.955894 

23.17 

23.10 

23.03 

22.95 

22.87 

22.82 

22.73 

22.65 

22.60 

22.52 

9.998220 

.18 

.20 

.18 

.20 

.18 

.20 

.20 

.18 

.20 

.20 

8.957674 

23.35 

23.30 

23.22 

23.15 

23.07 

23.00 

22.92 

22.87 

22.78 

22.72 

11.042326 

49 

12 

.957284 

.998209 

.959075 

.040925 

48 

13 

.958670 

.998197 

.960473 

.039527 

47 

14 

.960052 

.998186 

.961866 

.038134 

4o 

15 

.961429 

.998174 

.963255 

. 036745 

4.3 

16 

.962801 

.998163 

.964639 

.035361 

44 

17 

.964170 

.998151 

.966019 

.033981 

43 

18 

.965534 

.998139 

.967394 

.032606 

42 

19 

.966893 

.998128 

.968766 

.031234 

41 

20 

.968249 

.998116 

.970133 

.029867 

40 

21 

8.969600 

22.45 

22.37 

22.32 

22.23 

22.18 

22.10 

22.03 

21.97 

21.90 

21.83 

9.998104 

.20 

.20 

.20 

.20 

.20 

.20 

.20 

.20 

.20 

.20 

8.971496 

22.65 

22.57 

22.52 

22.43 

22.37 

22.30 

22.25 

22.17 

22.10 

22.03 

11.028504 

39 

22 

.970947 

.998092 

.972855 

.027145 

38 

23 

.972289 

.998080 

.974209 

.025791 

37 

24 

.973628 

.998068 

. 975560 

.024440 

36 

25 

.974962 

.998056 

.976906 

.023094 

35 

26 

.976293 

.998044 

.978248 

.021752 

34 

27 

.977619 

.998032 

.979586 

.020414 

33 

28 

.978941 

.998020- 

.980921 

.019079 

32 

29 

.980259 

.998008 

.982251 

.017749 

31 

30 

.981573 

.997996 

.983577 

.016423 

30 

31 

8.982883 

21.77 

21.72 

21.63 

21.57 

21.52 

21.43 

21.38 

21.32 

21.25 

21.18 

9.997984 

.20 

.22 

.20 

.20 

.22 

.20 

.22 

.20 

.22 

.20 

8.984899 

21.97 

21.92 

21.83 

21.78 

21.70 

21.65 

21.58 

21.53 

21.45 

21.40 

11.015101 

29 

32 

.984189 

.997972 

.986217 

.013783 

28 

33 

.985491 

.997959 

.987532 

.012468 

27 

34 

.986789 

.997947 

.988842 

.011158 

26 

35 

.988083 

.997935 

.990149 

.009851 

25 

36 

.989374 

.997922 

.991451 

.008549 

24 

37 

.990660 

.997910 

.992750 

.007250 

23 

38 

.991943 

.997897 

.994045 

.005955 

22 

39 

.993222 

.997885 

.995337 

.004663 

21 

40 

.994497 

.997872 

.996624 

.003376 

20 

41 

8.995768 

21.13 

21.05 

,21.02 

20.93 

20.88 

20.82 

20.75 

20.70 

20.65 

20.57 

9.997860 

.22 

.20 

.22 

.22 

.20 

.22 

.22 

22 

.22 

.22 

8.997908 

21.33 

21.28 

21.22 

21.15 

21.08 

21.03 

20.97 

20.92 

20.85 

20.80 

11.002092 

19 

42 

.997036 

.997847 

8.999188 

11.000812 

18 

43 

.998299 

.997835 

9.000465 

10.999535 

17 

44 

8.999560 

.997822 

.001738 

- .998262 

16 

45 

9.000816 

.997809 

.003007 

.996993 

15 

46 

.002069 

.997797 

.004272 

.995728 

14 

47 

.003318 

.997784 

.005534 

.994466 

13 

48 

.004563 

.997771 

.006792 

.993208 

12 

49 

.005805 

.997758 

.008047 

.991953 

11 

50 

.007044 

.997745 

* .009298 

.990702 

10 

51 

9.008278 

20.53 

20.45 

20.42 

20.33 

20.30 

20.22 

20.18 

20.12 

20.07 

9.997732 

- .22 
.22 
.22 
.22 
.22 
.22 
.22 
.22 
.23  | 

9.010546 

20.73 

20.68 

20.62 

20.57 

20.50 

20.45 

20.40 

20.33 

20.28 

10.989454 

9 

52 

.009510 

.997719 

.011790 

.988210 

8 

53 

.010737 

.997706 

.013031 

.986969 

7 

54 

.011962 

.997693 

.014268 

.985732 

6 

55 

.013182 

.997680 

.015502 

.984498 

5 

56 

.014400 

.997667 

.016732 

.983268 

4 

57 

.015613 

.997654 

.017959 

.982041 

3 

58 

.016824 

.997641 

.019183 

.980817 

2 

59 

.018031 

j .997628 

.020403 

.979597 

1 

60 

9.019235 

9.997614 

9.021620 

10.978380 

0 

' 

Cosine. 

D.  1". 

Sine. 

D.  1\  j 

| Cotang. 

1 D.l\ 

Tang. 

/ 

95< 


203 


84‘ 


6< 


TABLE  XII.  LOGARITHMIC  SIKES. 


173d 


' 

Sine. 

D.  1\ 

Cosine. 

D.  r. 

Tang. 

d.  r. 

! 

Cotang. 

/ 

0 

1 

2 

3 

4 

5 

6 

8 

9 

10 

11 

12 

13 

14 

15 

16 

17 

18 

19 

20 

21 

22 

23 

24 

25 

26 

27 

28 

29 

30 

31 

32 

33 

34 

35 

36 
37' 
33 

39 

40 

41 

42 

43 

44 

45 

46 

47 

48 

49 

50 

51 

52 

53 

54 

55 

56 

57 

58 

59 

60 

9.019235 

.020435 

.021632 

.022825 

.024016 

.025203 

.026386 

.027567 

.028744 

.029918 

.031089 

9.032257 

.03:3421 

.034582 

.035741 

.036896 

.038048 

.039197 

.040342 

.041485 

.042625 

9.043762 

.044895 

.046026 

.047154 

.048279 

.049400 

.050519 

.051635 

.052749 

.053859 

9.054966 

.056071 

.057172 

.058271 

.059367 

.060460 

.061551 

.062639 

.063724 

.064806 

9.065885 

.066962 

.068036 

.069107 

.070176 

.071242 

.072306 

.073366 

.074424 

.075480 

9.076533 

.077583 

.078631 

.079676 

.080719 

.081759 

.082797 

.083832 

.084864 

9.085894 

20.00 

19.95 

19.88 

19.85 
19.78 

19.72 
-19.68 

19.62 

19.57 
19  52 

19.47 

19.40 

19.35 

19.32 

19.25 

19.20 
19.15 

19.08 
19.05 
19.00 

18.95 

18.88 

18.85 
18.80 
18.75 
18.68 
18.65 
18.60 

18.57 

18.50 
18.45 

18.42 

18.35 

18.32 
18.27 
18.22 
18.18- 
18.13 

18.08 
18.03 
17.98 

17.95 
17.90 

17.85 
17.82 
17.77 

17.73 
17.67 

17.63 
17.60 
17.55 

17.50 

17.47  ! 

17.42  i 
17.38  1 

17.33 
17.30 

17.25 

17.20 
17.17 

j 9.997614 
.997601 
.997588 
.997574 
.997561 
.997547 
.997534 
.997520 
.997507 
.997493 
.997480 

9.997466 

.997452 

.997439 

.997425 

.997411 

.997397 

.997383 

.997369 

.997355 

.997341 

9.997327 

.997313 

.997299 

.997285 

.997271 

.997257 

.997242 

.997228 

.997214 

.997199 

9.997185 

.997170 

.997156 

.997141 

.997127 

.997112 

.997098 

.997083 

.997068 

.997053 

9.997039 

.997024 

.997009 

.996994 

.996979 

.996964 

.996949 

.996934 

.996919 

.996904 

9.996889 
.996874 
.996858 
.996843 
.996828 
.996812 
.996797 
. 996782 
.996766 
9.996751 

.22 

.22 

.23 

.22 

.23 

.22 

.23 

.22 

.23 

.22 

.23 

.23 

.22 

.23 

.23 

.23 

.23 

.23 

.23 

.23 

.23 

.23 

.23 

.23 

.23 

.23 

.25 

.23 

.23 

.25 

.23 

.25 

.23 

.25 

.23 

.25 

.23 

.25 

.25 

.25 

.23 

.25 

.25 

.25 

.25 

.25 

.25 

.25 

.25 

.25 

.25 

.25 

.27 

.25 

.27 

.27 

.25 

.25 

.27 

.25 

9.021620 
.022834 
.024044 
.025251 
. 026455 
.027655 
.028852 
.030046 
.031237 
.032425 
.033609 

9.034791 

.035969 

.037144 

.038316 

.039485 

.040651 

.041813 

.042973 

.044130 

.045284 

9.046434 

.047582 

.048727 

.049869 

.051008 

.052144 

.053277 

.054407 

.055535 

.056659 

9.057781 

.058900 

.060016 

.061130 

.062240 

.063348 

.064453 

.065556 

.066655 

.067752 

9.068846 
.069938 
.071027 
.072113 
.073197 
.074278 
. 075356 
.076432 
.077505 
. 078576 

9.079644 

.080710 

.081773 

.082833 

.083891 

.084947 

.086000 

.087050 

.088098 

9.089144 

20.23 

20.17 
20.12 

20.07 
20.00 
19.95 
19.90 
19.85 

19.80 

19.73 

19.70 

19.63 

19.58 

19.53 

19.48 

19.43 

19.37 
19.33 

19.28 

19.23 

19.17 

19.13 

19.08 
19.03 
18.98 

18.93 
18.88 
18  83 

18.80 

18.73 

18.70 

18.65 

18.60 

18.57 

18.50 

18.47 

18.42 

18.38 
18.32 

18.28 
*18.25 

18.20 

18.15 

18.10 

18.07 

18.02 

17.97 

17.93 
17.88 
17.85 
17.80 

17.77 

17.72 

17.67 

17.63 

17.60 

17.55 

17.50 

17.47 

17.43 

10.978380 

.977166 

.975956 

.974749 

.973545 

.972345 

.971148 

.969954 

.968763 

.967575 

.966391 

10.965209 

.964031 

.962856 

.961684 

.960515 

.959349 

.958187 

.957027 

.955870 

.954716 

10.953566 

.952418 

.951273 

.950131 

.948992 

.947856 

.946723 

.945593 

.944465 

.943341 

10.942219 
.941100 
.939984 
. 93887'0 
. 937760 
.936652 
.935547 
.934444 
.933345 
.932248 

10.931154 

.930062 

.928973 

.927887 

.926803 

.925722 

.924644 

.923568 

.922495 

.921424 

10.920356 
.919290 
.918227 
.917167 
.916109 
.915053 
.914000 
.912950 
.91 1902 

10.910856 

60 

59 

58 

57 

56 

55 

54 

53 

52 

51 

50 

49 

48 

47 

46 

45 

44 

43 

42 

41 

40 

39 

38 

37 

36 

35 

34 

33 

32 

31 

30 

29 

28 

27 

26 

25 

24 

23 

22 

21 

20 

19 

18 

17 

16 

15 

14 

13 

12 

11 

10 

9 

8 

7 

6 

5 

4 

3 

2 

1 

0 

/ I 

Cosine,  | 

d.  r.  j 

Sine. 

D.  1". 

Cotang. 

IxrTT 

Tang. 

96‘ 


204 


83" 


COSINES,  TANGENTS,  AND  COTANGENTS. 


172* 


/ 

Sine. 

D.  1\ 

Cosine. 

D.  1". 

Tang. 

D.  1".  1 

Cotang. 

' 

0 

9.085894 

17.13 

17.08 

17.05 

17.00 

16.97 

16.93 

16.88 

16.83 

16.82 

16.77 

16.72 

9.996751 

.27 

.25 

.27 

.27 

.25 

.27 

.27 

.27 

.25 

.27 

.27 

9.089144 

17.38 

17.35 

17.30 

17.27 

17.23 

17.18 

17.13 

17.12 

17.07 

17.03 

16.98 

10.910856 

60 

1 

.086922 

.996735 

.090187 

.909813 

59 

2 

.087947 

.996720 

.091228 

.908772 

58 

3 

.088970 

.996704 

.092266 

.907734 

57 

4 

.089990 

.996688 

.093302 

.906698 

56 

5 

.091008 

.996673 

.094336 

.905664 

55 

6 

.092024 

.996657 

.095367 

.904633 

54 

7 

.093037 

.996641 

.096395 

.903605 

53 

8 

.094047 

.996625 

.097422 

.902578 

5° 

9 

.095056 

.996610 

.098446 

.901554 

51 

10 

.096062 

.996594 

.099468 

.900532 

50 

11 

9.097065 

16.68 

16.65 

16.62 

16.57 

16.53 

16.48 

16.47 

16.42 

16.37 

16.35 

9.996578 

.27 

.27 

.27 

.27 

.27 

.27 

.28 

.27 

.27 

.27 

9.100487 

16.95 

16.92 

16.88 

16.83 

16.80 

16.77 

16.72 

16.68 

16.65 

16.62 

10.899513 

49 

12 

.098066 

.996562 

.101504 

.898496 

48 

13 

.099065 

.996546 

.102519 

.897481 

47 

14 

.100062 

.996530 

.103532 

.896468 

46 

15 

.101056 

.996514 

.104542 

.895458 

45 

16 

.102048 

.996498 

.105550 

.894450 

44 

17 

.103037 

.996482 

.106556 

.893444 

43 

18 

.104025 

.996465 

.107559 

.892441 

42 

19 

.105010 

.996449 

.108560 

.891440 

41 

20 

.105992 

.996433 

.109559 

.890441 

40 

21 

9.106973 

16.30 

16.27 

16.23 

16.20 

16.15 

16.12 

16.08 

16.05 

16.02 

15.97 

9.996417 

.28 

.27 

.27 

.28 

.27 

.28 

.27 

.28 

.27 

.28 

9.110556 

16.58 

16.53 

16.50 

16.47 

16.43 

16.40 

16.35 

16.33 

16.28 

16.25 

10.889444 

39 

22 

.107951 

.996400 

.111551 

.888449 

38 

23 

.108927 

.996384 

.112543 

.887457 

37 

24 

.109901 

.996368 

.113533 

.886467 

36 

25 

.110873 

.996351 

.114521- 

.885479 

35 

26 

.111842 

.996335 

.115507 

.884493 

34 

27 

.112809 

.996318 

.116491 

.883509 

33 

28 

.113774 

.996302 

.117472 

.882528 

32 

29 

.114737 

.996285 

.118452 

.881548 

31 

30 

.115698 

.996269 

.119429 

.880571 

30 

31 

9.116656 

15.95 

15.90 

15.87 

15.83 

15.80 

15.75 

15.73 

15.70 

15.65 

15.63 

9.996252 

.28 

.27 

.28 

.28 

.28 

■ .28 
.28 

9.120404 

16.22 

16.18 

16.15 

16.12 

16.08 

16.03 

16.02 

15.97 

15.95 

15.90 

10.879596 

29 

32 

.117613 

.996235 

.121377 

.878623 

28 

33 

.118567 

.996219 

.122348 

. 877652 

27 

34 

.119519 

.996202 

.123317 

.876683 

26 

35 

.120469 

.996185 

.124284 

.875716 

25 

36 

.121417 

.996168 

.125249 

.874751 

24 

37 

.122362 

.996151 

.126211 

.873789 

23 

38 

.123306 

.996134 

.127172 

.872828 

22 

39 

.124248 

.•996117 

.28 

.28 

.28 

.128130 

.871870 

21 

40 

.125187 

.996100 

.129087 

.870913 

20 

41 

9.126125 

15.58 

15.55 

15.53 

15.48 

15.45 

15.42 

15.40 

15.35 

15.32 

15.28 

9.996083 

.28 

.28 

.28 

.28 

.28 

.30 

.28 

.28 

.30 

.28 

9.130041 

15.88 

15.83 

15.82 

15.77 

15.75 

15.70 

15.68 

15.63 

15.62 

15.57 

10.868959 

19 

42 

.127060 

.996066 

. 130994 

.869006 

18 

43 

.127993 

.996049 

.131944 

.868056 

17 

44 

.128925 

.996032 

.132893 

.867107 

16 

45 

.129854 

.996015 

.133839 

.866161 

15 

46 

.130781 

.995998 

.134784 

.865216 

14 

47 

.131706 

.995980 

.135726 

.864274 

13 

48 

.132630 

.995963 

.136667 

.863333 

12 

49 

* .133551 

.995946 

.137605 

.862395 

11 

50 

.134470 

.995928 

• .138542 

.861458 

10 

51 

9.135387 

15.27 

15.22 

15.20 

15.15 

15.12 

15.10 

15.07 

15.02 

15.00 

9.995911 

.28 

.30 

.28 

.30 

.30 

.28 

.30 

.28 

30 

9.139476 

15.55 

15.52 

15.48 

15.45 

15.42 

15.38 

15.37 

15.32 

15.30 

10.860524 

9 

52 

.136303 

.995894 

.140409 

.859591 

8 

53 

.137216 

.995876 

.141340 

.858660 

7 

54 

.138128 

.995859 

.142269 

.857731 

6 

55 

.139037 

.995841 

.143196 

.856804 

5 

56 

.139944 

.995823 

.144121 

.855879 

4 

57 

.140850 

.995806 

.145044 

.854956 

3 

58 

.141754 

.995788 

' . 145966 

.854034 

2 

59 

.142655 

.995771 

.146885 

.853115 

1 

60 

9.143555 

9.995753 

9.147803 

10.852197 

0 

/ 

Cosine. 

1 D.  1\ 

! Sine. 

1 J).  r. 

Cotang. 

D.  1”. 

Tang. 

' 

97’ 


205 


8i 


8‘ 


TABLE  XII.  LOGARITH  M TO  SINES, 


171® 


/ 

Sine. 

D.  1". 

Cosine. 

D.  V. 

Tang. 

D.  1". 

Cotang. 

! 

0 

9.143555 

14.97 

14.93 

14.90 

14.88 

14.83 

14.82 

14.78 

14.73 

14.72 

14.70 

14.65 

9.995753 

.30 

.30 

.30 

.30 

.28 

.30 

.30 

.30 

.32 

.30 

.30 

9.147803 

15.25 

15.23 

15.20 

15.17 

15.15 

15.10 

15.08 

15.05 

15.02 

14.98 

14.97 

10.852197 

60 

1 

. 144453 

.995735 

.148718 

.851282 

59 

2 

.145349 

.995717 

.149632 

.850368 

58 

3 

. 146243 

.995699 

.150544 

.849456 

57 

4 

.147136 

.995681 

.151454 

.848546 

56 

5 

. 148026 

.995664 

. 152363 

.847637 

55 

6 

.148915 

.995646 

.153269 

.846731 

54 

7 

.149802 

.995628 

.154174 

.845826 

53 

8 

. 150686 

.995610 

.155077 

.844923 

52 

9 

.151569 

.995591 

.155978 

.844022 

51 

10 

.152451 

. 995573 

.156877 

.843123 

50 

11 

9. 153330 

14.63 

14.58 

14.57 

14.55 

14.50 

14.48 

14.43 

14.43 

14.38 

14.35 

9.995555 

.30 

9.157775 

14.93 

14.90 

14.87 

14.83 

14.82 

14.78 

14.75 

14.73 

14.70 

14.67 

10.842225 

49 

12 

.154208 

.995537 

.158671 

.841329 

48 

13 

. 155083 

.995519 

.30 

.159565 

.840435 

47 

14 

. 155957 

.995501 

.160457 

.839543 

46 

15 

. 156830 

.995482 

.30 

.30 

.32 

.30 

.32 

.30 

.161347 

.838653 

45 

16 

.157700 

.995464 

, .162236 

.837764 

44 

17 

.158569 

.995446 

.163123 

.836877 

43 

18 

. 159435 

.995427 

.164008 

.835992 

42 

19 

. 160301 

.995409 

.164892 

.835108 

41 

20 

.161164 

.995390 

.165774 

.834226 

40 

21 

9.162025 

14.33 

14.30 

14.28 

14.23 

14.22 

14.20 

14.15 

14.13 

14.10 

14.08 

9.995372 

.32 

.32 

.30 

.32 

.32 

.30 

.32 

.32 

.32 

.32 

9.166654 

14.63 

14.62 

14.58 

14.55 

14.53 

14.50 

14.47 

14.45 

14.42 

14.38 

10.833346 

39 

22 

. 162885 

.995353 

.167532 

.832468 

38 

23 

.163743 

.995334 

. 168409 

.831591 

37 

24 

.164600 

.995316 

.169284 

.830716 

36 

25 

.165454 

.995297 

.170157 

.829843 

35 

26 

.166307 

. 995278 

.171029 

.828971 

34 

27 

.167159 

.995260 

.171899 

.828101 

33 

28 

.168008 

.995241 

. 172767 

.827233 

32 

29 

.168856 

.995222 

.173634 

.826366 

31 

30 

.169702 

.995203 

. 174499 

.825501 

30 

31 

9.170547 

14.03 

14.02 

14.00 

13.97 

13.93 

13.90 

13.88 

13.85 

13.83 

13.80 

9.995184 

09 

9.175362 

14.37 
14.33 
14.30 
14.28 
14  27 
14.22 
14.20 
14.18 
14.13 
14.13 

10.824638 

29 

32 

.171389 

.995165 

.32 

.32 

.32 

.32 

.32 

.32 

.32 

.32 

.33 

.176224 

.823776 

28 

33 

.172230 

. 995146 

.177084 

.822916 

27 

34 

.173070 

.995127 

.177942 

.822058 

26 

35 

.173908 

.995108 

.178799 

.821201 

25 

36 

.174744 

.995089 

.179655 

.820345 

24 

37 

. 175578 

.995070 

.180508 

.819492 

I 23 

38 

.176411 

. 995051 

.181360 

.818640 

22 

39 

. 177242 

.995032  , 

.182211 

.817789 

l 21 

40 

.178072 

.995013 

.183059 

.816941 

20 

41 

9.178900 

13.77 

13.75 

13.72 

13.70 

13.67 

13.63 

13.62 

13.58 

13.57 

13.53 

9.994993 

.32 

.32 

33 

9.183907 

14.08 

14.08 

14.03 

14.02 

14.00 

13.97 

13.93 

13.92 

13.88 

13.87 

10.816093 

19 

42 

. 179726 

.994974 

.184752 

.815248 

18 

43 

.180551 

.994955 

.185597 

.814403 

17  . 

44 

.181374 

.994935 

i32 

.33 

.32 

.33 

.32 

.33 

.33 

.186439 

.813561 

16 

45 

.182196 

.994916 

.187280 

.812720 

15 

46 

.183016 

.994896 

.188120 

.811880 

14 

47 

.183834 

.994877 

.188958 

.811042 

13 

48 

. 184651 

.994857 

.189794  . 

.810206 

12 

49 

.185466 

- .994838 

.190629 

.809371 

11 

50 

.186280 

.994818 

.191462 

.808538 

10 

51 

9.187092 

13.52 

13.48 

13.45 

13.43 

13.42 

13.38 

13.35 

13.33 

13.30 

9.994798 

.32 

.33 

.33 

.32 

.33 

.33 

.33 

.33 

.33 

9.192294 

13.83 

13.82 

13.78 

13.77 

13.73 

13.72 

13.68 

13.67 

13.65 

10.807706 

9 

52 

.187903 

.994779 

.193124 

.806876 

8 

53 

. 188712 

.994759 

.193953 

.806047 

7 

54 

.189519 

.994739 

.194780 

'.805220 

6 

55 

.190325 

.994720 

.195606 

.804394 

5 

56 

.191130 

.994700 

.196430 

.803570 

4 

57 

.191933 

.994680 

.197253 

.802747 

3 

58 

.192734 

.994660 

.198074 

.80192(5 

2 

59 

.193534 

.994640 

.198894 

.801106 

1 

60 

9.194332 

9.994620 

9.199713 

10.800287 

C 

Cosine. 

D.  1".  1 

Sine. 

D.  1".  1 

Cotang.  | 

D.  1". 

Tang. 

/ 1 

98< 


206 


81 


COSINES,  TANGENTS,  AND  COTANGENTS. 


170‘ 


/ 

Sine. 

t 

D.  1\  ! 

Cosine. 

D.  1". 

Tang. 

D.  1M. 

Cotang. 

/ 

0 

9.194332 

13.28  1 

13.27 

13.23 

13.20 

13.18 

13.15 

13.13 

13.12 

13.08 

13.05 

13.05 

9.994620 

.33 

.33 

.33 

.33 

.35 

.33 

.33 

.33 

.35 

.33 

.33 

9.199713 

13.60 

13.60 

13.57 

13.53 

13.52 

13.50 

13.47 

13.45 

13.43 

13.40 

13.37 

10.800287 

60 

1 

.195129 

.994600 

.200529 

.799471 

59 

2 

.195925 

.994580 

.201345 

.798655 

58 

3 

.196719 

.994560 

.202159 

.797841 

57 

4 

.197511 

.994540 

.202971 

.797029 

56 

5 

.198302 

.994519 

.203782 

.796218 

55 

6 

.199091 

.994499 

.204592 

.795408 

54 

7 

.199879 

.994479 

.205400 

.794600 

53 

8 

.200666 

.994459 

.206207 

.793793 

52 

9 

.201451 

.994438 

.207013 

.792987 

51 

10 

.202234 

.994418 

.207817 

.792183 

50 

11 

9.203017 

13.00 

13.00 

12.95 

12.95 

12.92 

12.88 

12.88 

12.83 

12.83 

12.80 

9.994398 

.35 

.33 

.35 

.33 

.35 

.35 

.33 

.35 

.35 

.35 

9.208619 

13.35 

13.33 

13.30 

13.28 

13.27 

13.23 

13.22 

13.18 

13.18 

13.13 

10.791381 

49 

12 

.203797 

.994377 

.209420 

.790580 

48 

13 

.204577 

.994357 

.210220 

.789780 

47 

14 

.205354 

.994336 

.211018 

.788982 

46 

15 

.206131 

.994316 

.211815 

.788185 

45 

16 

.206906 

.994295 

.212611 

.787389 

44 

17 

.207679 

.994274 

.213405 

.786595 

43 

18 

.208452 

.994254 

.214198 

.785802 

42 

19 

.209222 

.994233 

.214989 

.785011 

41 

20 

.209992 

.994212 

.215780 

.784220 

40 

21. 

9.210760 

12.77 

12.75 

12.73 

12.72 

12.68 

12.65 

12.65 

12.62 

12.58 

12.57 

9.994191 

.33 

.35 

.35 

.35 

.35 

.35 

.35 

.35 

.35 

.35 

9.216568 

13.13 

'13.10 

13.07 

13.07 

13.03 

13.00 

13.00 

12.97 

12.95 

12.92 

10.783432 

39 

22 

.211526 

.994171 

.217356 

.782644 

38 

23 

.212291 

.994150 

.218142 

.781858 

37 

24 

.213055 

.994129 

.218926 

.781074 

36 

25 

.213818 

.994108 

.219710 

.780290 

35 

26 

.214579 

.994087 

.220492 

.779508 

34 

27 

.215338 

.994066 

.221272 

.778728 

33 

28 

.216097 

.994045 

.222052 

.777948 

32 

29 

.216854 

.994024 

.222830 

.777170 

31 

30 

.217609 

.994003 

.223607 

.776393 

30 

31 

9.218363 

12.55 

12.53 

12.50 

12.48 

12.47 

12.43 

12.42 

12.38 

12.38 

12.35 

9.993982 

.37 

.35 

.35 

.35 

.37 

.35 

.37 

.35 

.37 

.35 

9.224382 

12.90 

12.88 

12.85 

12.85 

12.80 

12.80 

12.77 

12.77 

12.72 

12.72 

10.775618 

29 

32 

.219116 

.993960 

.225156 

.774844 

28 

33 

.219868 

.993939 

.225929 

.774071 

27 

34 

.220618 

.993918 

.226700 

.773300 

26 

35 

.221367 

.993897 

.227471 

.772529 

25 

36 

.222115 

.993875 

.228239 

.771761 

24 

37 

.222861 

.993854 

.229007 

.770993 

23 

38 

.223606 

.993832 

.229773 

.770227 

22 

39 

.224349 

.993811 

.230539 

.769461 

21 

40 

.225092 

.993789 

.231302 

.768698 

20 

41 

9.225833 

12.33 

12.30 

12.28 

12.27 

12.23 

12.23 

12.20 

12.18 

12.15 

12.13 

9.993768 

.37 

.35 

.37 

.37 

.35 

.37 

.37  ' 

.37 

.37 

.37 

9.232065 

12.68 

12.67 

12.65 

12.63 

12.60 

12.58 

12.57 

12.53 

12.53 

12.50 

10.767935 

19 

42 

.226573 

.993746 

.232826 

.767174 

18 

43 

.227311 

.993725 

.233586 

.766414 

17 

44 

.228048 

.993703 

.234345 

.765655 

16 

45 

.228784 

.993681 

.235103 

.764897 

15 

46 

.229518 

.993660 

.235859 

.764141 

14 

47 

.230252 

.993638 

.236614 

. 763386 

13 

48 

.230984 

.993616 

.237368 

.762632 

12 

49 

.231715 

.993594 

.238120 

.761880 

11 

50 

.232444 

.993572 

• .238872 

.761128 

10 

51 

9.233172 

12.12 

12.10 

12.07 

12.07 

12.03 

12.00 

12.00 

11.97 

11.95 

9.993550 

.37 
.37 
.37 
• .37 
.37 
.37 
.37 
.37 

QQ 

9.239622 

12.48 

12.45 

12.45 

12.42 

12.40 

12.38 

12.37 

12.33 

12.33 

10.760378 

9 

52 

.233899 

.993528 

.240371 

.759629 

8 

53 

.234625 

.993506 

.241118 

.758882 

7 

54 

.235349 

.993484 

.241865 

.758135 

6 

55 

.236073 

.993462 

.242610 

.757390 

5 

56 

1 .236795 

.'993440 

.243354 

.756646 

4 

57 

.237515 

.993418 

.244097 

.755903 

3 

58 

.238235 

.993396 

.244839 

.755161 

2 

59 

.238953 

.993374 

.245579 

.754421 

1 

60 

9.239670 

9.993351 

.OO 

9.246319 

10.753681 

0 

7 

Cosine. 

1 D.  1".  J 

Sine.  * 

D.  1".  i 

Cotang. 

D.  1". 

Tang. 

7 

89' 


207 


80' 


10' 


TABLE  XII.  LOGARITHMIC  SINES, 


169" 


' 

Sine. 

D.  1". 

Cosine. 

D.  1". 

Tang. 

D.  1\ 

Cotang.  | 

' 

0 

9.239670 

11.93 

11.92 

11.88 

11.87 

11.85 

11.83 

11.82 

11.78 

11.77 

11.77 

11.72 

9.993351 

.37 
.37  S 

9.246319 

12.30 
12.28 
12.27 
12.23 
12  23 
12.20 
12.18 
12.17 
12.15 
12.13. 
12.10 

10.753681 

60 

1 

.240386 

.993329 

.247057 

.752943 

59 

2 

.241101 

.993307 

.247794 

.752206 

i 58 

3 

.241814 

.993284 

.OO 

.37 

.37 

.38 

.37 

.248530 

.751470 

57 

4 

.242526 

.993262 

.249264 

.750736 

56 

5 

.243237 

.993240 

.249998 

.750002 

55 

6 

.243947 

.993217 

.250730 

.749270 

54 

7 

.244656 

.993195 

.251461 

.748539 

53 

8 

.245363 

.993172 

.do 

.38 

.37 

.38 

.252191 

.747809 

52 

9 

.246069 

.993149 

.252920 

.747080 

51 

10 

.246775 

.993127 

.253648 

.746352 

50 

11 

9.247478 

11.72 

11.70 

11.67 

11.65 

11.63 

11.62 

11.60 

11.57 

11.57 

11.53 

9.993104 

.38 

.37 

.38 

.38 

.38 

.38 

9.254374 

12.10 

12.07 

12.05 

12.03 

12.02 

12.00 

11.98 

11.95 

11.95 

11.92 

10.745626 

49 

12 

.248181 

.993081 

.255100 

.744900 

48 

13 

.248883 

.993059 

.255824 

.744176 

47 

14 

.249583 

.993036 

.256547 

.743453 

46 

15 

.250282 

.993013 

.257269 

.742731 

45 

16 

.250980 

.992990 

.257990 

.742010 

44 

17 

.251677 

.992967 

.258710 

.741290 

43 

18 

.252373 

.992944 

.dO 

.38 

.38 

.38 

.259429 

.740571 

42 

19 

.253067 

.992921 

.260146 

.739854 

41 

20 

.253761 

.992898 

.260863 

.739137 

40 

21 

9.254453 

11.52 

11.50 

11.48 

11.47 

11.45 

11.42 

11.42 

11.38 

11.37 

11.35 

9.992875 

.38 

9.261578 

11.90 

11.88 

11.87 

11.85 

11.83 

11.82 

11.80 

11.77 

11.77 

11.73 

10.738422 

39 

22 

.255144 

.992852 

.262292 

.737708 

38 

23 

.255834 

.992829 

.38  ; 

.263005 

.736995 

37 

24 

.256523 

.992806 

.38 

.38 

.40 

.38 

.38 

.263717 

.736283 

36 

25 

.257211 

.992783 

.264428 

.735572 

35 

26 

.257898 

.992759 

.265138 

.734862 

34 

27 

.258583 

.992736 

.265847 

.734153 

33 

28 

.259268 

.992713 

.266555 

.733445 

32 

29 

.259951 

.992690 

.38 

.40 

.38 

.267261 

.732739 

31 

30 

.260633 

.992666 

.267967 

.732033 

30 

31 

9.261314 

11.33 

11.32 

11.30 

11.27 

11.27 

11.23 

11.23 

11.20 

11.20 

11.17 

9.992643 

.40 

.38 

.40 

qq 

9.268671 

11.73 
11.70 
11.70 
• 11.67 
11.65 
11.63 
11.62 
11.60 
11.58 
11.57 

10.731329 

29 

32 

.261994 

.992619 

.269375 

.730625 

28 

33 

.262673 

.992596 

.270077 

.729923 

27 

34 

.263351 

.992572 

.270779 

.729221 

26 

35 

.264027 

.992549 

.OO 

.40 

.40 

qq 

.271479 

.728521 

25 

36 

.264703 

.992525 

.272178 

.727822 

24 

37 

.265377 

.992501 

.272876 

.727124 

23 

38 

.266051 

.992478 

.OO 

.40 

.40 

.40 

.273573 

.726427 

22 

39 

.266723 

.992454 

.274269 

.725731 

21 

40 

.267395 

.992430 

.274964 

.725036 

20 

41 

9.268065 

11.15 
11.13 
11.12 
11.10 
11.08 
11.07 
11.03 
11.03 
11.02 
40  98 

9.992406 

.40 

.38 

.40 

.40 

.40 

.40 

.40 

.42 

.40 

.40 

9.275658 

11.55 

11.53 

11.52 

11.50 

11.48 

11.47 

11.45 

11.43 

11.40 

11.40 

10.724342 

19 

42 

.268734 

.992382 

.276351 

.723649 

18 

43 

.269402 

.992359 

.277043 

.722957 

17 

44 

.270069 

.992335 

.277734 

.722266 

16 

45 

.270735 

.992311 

.278424 

.721576 

15 

46 

.271400 

.992287 

.279113 

.720887 

14 

47 

.272064 

. 992263 

.279801 

.720199 

13 

48 

.272726 

.992239 

.280488 

.719512 

12 

49 

.273388 

.992214 

.281174 

.718826 

11 

60 

.274049 

.992190 

.281858 

.718142 

10 

51 

9.274708 

10.98 

10.97 

10.93 

10.93 

10.90 

10.90 

10.87 

10.85 

10.85 

9.992166 

.40 
.40 
.42 
.40 
.42' 
.40 
.40 
.42 
.40  i 

9.282542 

11.38 

11.37 

11.35 

11.33 

11.32 

11.28 

11.28 

11.27 

11.25 

10.717458 

9 

52 

.275367 

.992142 

.283225 

.716775 

8 

53 

.276025 

.992118 

.283907 

.716093 

7 

54 

.276681 

.992093 

.284588 

.715412 

C 

55 

.277337 

.992069 

.285268 

.714732 

5 

56 

.277991 

.992044 

.285947 

.714053 

4 

57 

.278645 

.992020 

.286624 

.713376 

3 

58 

279297 

.991996 

.287301 

.712699 

2 

59 

.279948 

.991971 

.287977 

.712023 

1 

60 

9.280599 

9.991947 

9.288652 

10.711348 

0 

/ 

Cosine. 

D.  r. 

1 Sine.  1 

D.  1".  1 

Cotang. 

D.  1\  1 

Tang.  | 

' 

100 


208 


79' 


COSINES.  TANGENTS,  AND  COTANGENTS. 

11°  9 9 1R8# 


/ 

Sine. 

d.  r.  ! 

Cosine. 

D.  1". 

1 

| Tang. 

D.  1". 

Cotang. 

/ 

0 

9.280599 

10.82 

10.82 

10.78 

10.77 

10.77 

10.73 

10.73 

10.70 

10.70 

10.67 

10.67 

9.991947 

.42 

.42  I 

.40  1 

.42 

.42  1 

.40 

.42 

.42 

.42 

.42 

.42 

9.288652 

11.23 

11.22 

11.20 

11.18 

11.18 

11.15 

11.13 

11.12 

11.12 

11.08 

11.07 

10.711348 

60 

1 

-.281248 

.991922 

.289326 

.710674 

59 

2 

.281897 

.991897 

.289999 

.710001 

58 

3 

.282544 

.991873 

.290671 

.709329 

57 

4 

.283190 

.991848 

.291342 

.708658 

56 

5 

.283836 

.991823 

.292013 

.707987 

55 

6 

.284480 

.991799 

.292682 

.707318 

54 

7 

.285124 

.991774 

.293350 

.706650 

53 

8 

.285766 

.991749 

.294017 

.705983 

52 

9 

.286408 

.991724 

.294684 

.705316 

51 

10 

.287048 

.991699 

.29^349 

.704651 

50 

11 

9.287688 

10.63 

10.63 

10.60 

10.60 

10.57 

10.57 

10.55 

10.52 

10.52 

10.50 

9.991674 

AT 

Ak 

.42 

.42 

.42 

.42 

.43 

.42 

.42 

.43 

9.296013 

11.07 

11.03 

11.03 

11.02 

11.00 

10.97 

10.97 

10.95 

10.93 

10.93 

10.703987 

49 

12 

.288326 

.991649 

.296677 

.703323 

48 

13 

.288964 

.991624 

.297339 

.702661 

47 

14 

.289600 

.991599 

.298001 

.701999 

46 

15 

.290236 

.991574 

.298662 

.701338 

45 

16 

.290870 

.991549 

.299322 

.700678 

44 

17 

.291504 

.991524 

.299980 

.700020 

43 

18 

.292137 

.991498 

.300638 

.699362 

42 

19 

.292768 

.991473 

.301295 

.698705 

41 

20 

.293399 

.991448 

.301951 

.698049 

40 

21 

9.294029 

10.48 

10.47 

10.45 

10.43 

10.42 

10.40 

10.40 

10.37 

10.35 

10.35 

9.991422 

.42 

.42 

.43 

.42 

.43 

.42 

.43 

.43 

.42 

.43 

9.302607 

10.90 

10.88 

10.88 

10.85 

10.85 

10.83 

10.82 

10.80 

10.78 

10.77 

10.697393 

39 

22 

.294658 

.991397 

.303261 

.696739 

38 

23 

.295286 

.991372 

.303914 

.696086 

37 

24 

.295913 

.991346 

.304567 

.695433 

36 

25 

.296539 

.991321 

.305218 

.694782 

35 

26 

.297164 

.991295 

.305869 

.694131 

34 

27 

.297788 

.991270 

.306519 

.693481 

33 

28 

.298412 

.991244 

.307168 

.692832. 

32 

29 

.299034 

.991218 

.307816 

.692184 

31 

30 

.299655 

.991193 

.308463 

.691537 

30 

31 

9.300276 

10.32 

10.32 

10.30 

10.27 

10.27 

10.25 

10.23 

10.23 

10.20 

10.18 

9.991167 

.43 

.43 

-.42 

.43 

.43 

.43 

.43 

.43 

.43 

.43 

9.309109 

10.75 

10.75 

10.72 

10.72 

10.70 

10.68 

10.67 

10.65 

10.63 

10.63 

10.690891 

29 

32 

.300895 

.991141 

.309754 

.690246 

28 

33 

.301514 

.991115 

.310399 

.689601 

27 

34 

.302132 

.991090 

.311042 

.688958 

26 

35 

.302748 

.991064 

.311685 

.688315 

25 

36 

.303364 

.991038 

.312327 

.687673 

24 

37 

.303979 

.991012 

.312968 

.687032 

23 

38 

.304593 

.990986 

.313608 

.686392 

22 

39 

.305207 

.990960 

.314247 

.685753 

21 

40 

.305819 

.990934 

.314885 

.685115 

20 

41 

9.308430 

10.18 

10.15 

10.15 

10.13 

'10.12 

10.10 

10.08 

10.07 

10.07 

10.03 

9.990908 

.43 

.45 

.43 

.43 

.43 

.45 

.43 

.45 

.43 

.43 

9.315523 

10.60 

10.60 

10.58 

10.57 

10.55 

10.55 

10.52 

10.52 

10.50 

10.48 

10.684477 

19 

42 

.307041 

.990882 

.316159 

.683841 

18 

43 

.307650 

.990855 

.316795 

.683205 

17 

44 

.308259 

.990829 

.317430 

.682570 

16 

45 

.308867 

.990803 

.318064 

.681936 

15 

46 

.309474 

.990777 

.318697 

.681303 

14 

47 

.310080 

.990750 

319330 

. 680670 

13 

48 

.310685 

.990724 

.319961 

.680039 

12 

49 

.311289 

.990697 

.320592 

.679408 

11 

50 

.311893 

.990671 

.321222 

.678778 

10 

51 

9.312495 

10.03 

10.02 

9.98 

10.00 

9.97 

9.95 

9.95 

9.92 

9.92 

9.990645 

.45 

.45 

.43 

.45 

.45 

.43 

.45 

.45 

.45 

9.321851 

10.47 

10.45 

10.45 

10.42 

10.42 

10.40 

10.40 

10.37 

10.37 

10.678149 

9 

52 

.313097 

.990618 

.322479 

.677521 

8 

53 

.313698 

.990591 

.323106 

.676894 

7 

54 

,314297 

.990565  ! 

.323733 

.676267 

6 

65 

.314897 

.990538  } 

.324358 

.675642 

5 

56 

.315495 

.990511 

.324983 

.675017 

4 

57 

.316092 

.990485 

.325607 

.674393 

3 

58 

.316689 

.990458 

.326231 

.673769 

2 

59 

.317284 

.990431 

.326853 

.673147 

1 

60 

9.317879 

9.990404 

9.327475 

10.672525 

0 

/ 

Cosine. 

D.  1*. 

i Sine. 

d.  r.  1 

Cotang,  i 

D.  1\  I 

Tang. 

/ 

101< 


209 


78’ 


12‘ 


TABLE  XIT.  LOGARITHMIC  SINES, 


167* 


' 

Sine. 

D.  r. 

Cosine. 

D.  1". 

Tang. 

D.  r. 

Cotang. 

/ 

0 

9.317879 

9.90 

9.88 

9.87 

9.85 

9.85 

9.83 

9.82 

9.80 

9.78 

9.77 

9.77 

9.73 

9.73 

9.72 

9.72 

9.68 

9.68 

9.67 

9.65 

9.63 

9.62 

9.62 

9.60 

9.57 

9.58 
9.55 
9.55 

9.52 

9.53 
9.50 
9.48 

9.48 

9.47 

9.45 

9.43 

9.43 

9.42 

9.40 

9.38 

9.37 

9.37 

9.35 

9.33 

9.33 

9.30 

9.30 

9.28 

9.28 

9.25 

9.25 

9.25 

9.22 

9.22 

9.20 

9.18 

9.17 

9.17 

9.15 

9.13 

9.13 

9.990404 

.43 

.45 

.45 

.45 

.45 

.45 

.47 

.45 

.45 

9.327475 

10.33 

10.33 

10.32 

10.32 

10.28 

10.28 

10.27 

10.25 

10.25 

10.22 

10.22 

10.20 

10.18 

10.18 

10.15 

10.15 

10.13 

10.13 

10.10 

10.10 

10.08 

10.07 

10.07 

10.05 

10.03 

10.02 

10.00 

10.00 

9.98 

9.97 

9.97 

9.93 

9.93 

9.93 

9.90 

9.90 

Q QQ 

10.672525 

60 

1 

.318473 

.990378 

.328095 

.671905 

59 

2 

.319066 

.990351 

.328715 

.671285 

58 

3 

.319658 

.990324 

.329334 

.670666 

57 

1 

.320249 

.99029? 

.329953 

.670047 

56 

5 

.320840 

.990270 

.330570 

.669430 

55 

6 

.321430 

.990243 

.331187 

.668813 

54 

7 

.322019 

.990215 

.331803 

.668197 

53 

8 

.322607 

.990188 

.332418 

.667582 

52 

9 

.323194 

.990161 

.333033 

.666967 

51 

10 

11 

.323780 

9.324366 

.990134 

9.990107 

.45 

.45 

.47 

.45 

.45 

.47 

.45 

.47 

.45 

.47 

.45 

.47 

• .47 

.45 
.47 
.47 
.47 

.333646 

9.334259 

.666354 

10.665741 

50 

49 

12 

.324950 

.990079 

.334871 

.665129 

48 

13 

.325534 

.990052 

.335482 

.664518 

47 

14 

. .326117 

.990025 

.336093 

.663907 

46 

15 

.326700 

.989997 

.336702 

.663298 

45 

1G 

.327281 

.989970 

.337311 

.662689 

44 

17 

.327862 

.989942 

.337919 

.662081 

43 

13 

.328442 

.989915 

.338527 

.661473 

42 

19 

.329021 

.989887 

.339133 

.660867 

41 

20 

21 

.329599 

9.330176 

.989860 

9.989832 

.339739 

9.340344 

.660261 

10.659656 

40 

39 

22 

.330753 

.989804 

.340948 

.659052 

38 

23 

.331329 

.989777 

' .341552 

.658448 

37 

24 

.331903 

.989749 

.342155 

.657845 

36 

25 

.332478 

.989721 

.342757 

.657243 

35 

2G 

.333051 

.989693 

.343358 

.656642 

34 

27 

.333624 

.989665 

.47 

.45 

.47 

.48 

.47 

.47 

.47 

.47 

.47 

.47 

.48 

.47 

.47 

.48 

.47 

.48 

.47 

.48 

.48 

.343958 

.656042 

33 

28 

.334195 

.989637 

i .344558 

.655442 

i 32 

29 

.334767 

.989610 

.345157 

.654843 

31 

30 

31 

.335337 

9.335906 

.989582 

9.989553 

.345755 

9.346353 

.654245 

10.653647 

30 

29 

32 

.336475 

.989525 

.346949 

.653051 

28 

33 

.337043 

.989497 

.347545 

.652455 

27 

34 

.337610 

.989469 

.348141 

.651859 

26 

35 

.338176 

.989441 

.348735 

.651265 

25 

36 

.338742 

.989413 

.349329 

.650671 

24 

37 

.339307 

.989385 

.349922 

y.oo 

9.87 

9.87 

9.85 

9.83 

9.82 

9.82 

9.80 

9.78 

9.78 

O 77 

.650078 

23 

38 

.339871 

.989356 

.350514 

.649486 

22 

39 

.340434 

.989328 

.351106 

.648894 

21 

40 

41 

.340996 

9.341558 

.989300 

9.989271 

.351697 

9.352287 

.648303 

10.647713 

20 

19 

42 

.342119 

.989243 

.352876 

.647124 

18 

43 

.342679 

.989214 

.353465 

.646535 

17 

44 

.343239 

.989186 

.354053 

.645947 

16 

45 

.343797 

.989157 

.354640 

.645360 

15 

46 

.344355 

.989128 

.355227 

.644773 

14 

47 

.344912 

.989100 

.47 

.48 

..40 

.47 

.48 

.48 

.48 

.48 

.48 

.48 

.48 

.48 

.48 

.48 

.355813 

Q 7\ 

.644187 

13 

48 

.345469 

.989071 

.356398 

y.  (O 

9.73 

9.73 

9.72 

9.70 

9.70 

9.67 

9.68 
9.65 
9.65 
9.63 
9.62  i 
9.62 

.643602 

12 

49 

.346024 

.989042 

.356982 

.643018 

11 

50 

51 

.346579 

9.347134 

.989014 

9.988985 

.357566 

9.358149 

.642434 

10.641851 

10 

9 

52 

.347687 

.988956 

.358731 

.641269 

8 

53 

.348240 

.988927 

.359313 

.640687 

7 

54 

.348792 

.988898 

.359893 

.640107 

6 

55 

.349343 

.988869 

.360474 

.639526 

5 

56 

.349893 

.988840 

.361053 

. 638947 

4 

57 

.350443 

.988811 

.361632 

.638368 

3 

58 

.350992 

.988782 

.362210 

.637790 

2 

59 

.351540 

.988753 

.362787 

.637213 

1 

60 

9.352088 

9.988724 

9.363364 

10.636636 

0 

/ 

Cosine. 

D.  1 . 

Sine, 

d.  r. 

1 Cotang. 

d.  r.  ! 

Tang. 

' 

COSTNES*  TANGENTS,  AND  COTANGENTS. 


166* 


13° 


' 

Sine. 

D.  1". 

Cosine.  ] 

D.  1". 

Tang. 

d.  r. 

Cotang. 

' 

0 

9.352088 

9.12 

9.10 

9.08 

9.08 

9.07 

9.05 

9.05 

9.03 

9.02 

9.00 

9.00 

9.988724 

.48 

9.363364 

9.60 

9.58 

9.58 

9.57 

9.55 

9.55 

9.53 

9.52 

9.52 

9.50 

9.48 

10.C3C636 

CO 

1 

.352635 

.988695 

.363940 

.636060 

59  ' 

2 

.353181 

.988666 

.50 

.48 

.48 

.50 

.48 

.50 

.48 

.50 

.48 

.364515 

. 635485 

58 

3 

.353726 

.988636 

.365090 

.634910 

57 

4 

.354271 

.988607 

.365664 

.634336 

50 

5 

.354815 

.988578 

.366237 

.633763 

55 

6 

.355358 

.988548 

.366810 

.633190 

54 

7 

.355901 

.988519 

.367382 

.632618 

53 

8 

.356443 

.988489 

.367953 

.632047 

52 

9 

.356984 

.988460 

.368524 

.631476 

51 

10 

.357524 

.988430 

.369094 

.630906 

50 

11 

9.358064 

8.98 

8.97 

8.95 

8.95 

8.95 

8.92 

8.92 

8.90 

8.88 

8.88 

9.988401 

.50 

.48 

.50 

.50 

.50 

.48 

.50 

.50 

.50 

.50 

9.369663 

9.48 

9.45 

9.47 

9.43 

9.43 

9.42 

9.42 

9.40 

9.38 

9.38 

10.630337 

49 

12 

.358603 

.988371 

.370232 

.629768 

48 

13 

.359141 

.988342 

.370799 

.629201 

47 

14 

. 359678 

.988312 

.371367 

.628633 

46 

15 

.360215 

.988282 

.371933 

.628067 

45  - 

16 

.360752 

.988252 

. 372499 

.627501 

44 

17 

.361287 

.988223 

.373064 

.626936 

43 

18 

.361822 

.988193 

.373629 

.626371 

42 

19 

.362356 

.988163 

.374193 

.625807 

41 

20 

.362889 

.988133 

.374756 

. C25244 

40 

21 

9.363422 

8.87 

8.85 

8.85 

8.83 

8.82 

8.82 

8.78 

8.80 

8.77 

8.77 

9.988103 

.50 

.50 

.50 

.50 

.50 

.52 

.50 

.50 

.50 

.52 

9.375319 

9.37 
9.35 
9. 35 
9.33 
9.32 
9.32 
9.30 
9.30 
9.28 
9.27 

10.624681 

39 

22 

.363954 

.988073 

.375881 

.624119 

38 

23 

.364485 

.988043 

.376442 

. 623558 

37 

24 

.365016 

.988013 

.377003 

.622997 

36 

25 

.365546 

.987983 

.377563 

.622437 

35 

26 

.366075 

.987953 

.378122 

.621878 

34 

27 

.366604 

.987922 

.378681 

.621319 

33 

28 

.367131 

.987892 

.379239 

.620761 

32 

29 

.367659 

.987862 

.379797 

.620203 

31 

30 

.368185 

.987832 

.380354 

.619646 

30 

31 

9.368711 

8.75 

8.75 

8.72 

8.72 

8.70 

8.70 

8.68 

8.68 

8.67 

8.65 

9.987801 

.50 

.52 

.50 

.52 

.50 

.52 

.50 

.52 

.52 

.50 

9.380910 

9.27 

9.23 

9.25 

9.23 

9.22 

9.20 

9.20 

9.18 

9.18 

9.17 

10.619090 

29 

32 

.369236 

.987771 

.381466 

.618534 

28 

33 

.369761 

.987740 

.382020 

.617980 

27 

34 

.370285 

.987710 

.382575 

.617425 

26 

35 

.370808 

.987679 

.383129 

.616871 

25 

36 

.371330 

.987649 

.383682 

.616318 

24 

37 

.371852 

.987618 

.384234 

.615766 

23 

38 

.372373 

.987588 

.384786 

.615214 

22 

39 

.372894 

.987557 

.385337 

.614663 

21 

40 

.373414 

.987526 

.385888 

.614112 

20 

41 

9.373933 

8.65 

8.63 

8.62 

,8.60 

8.60 

8.60 

8.57 

8.57 

8.57 

8.53 

9.987496 

.52 

.52 

.52 

.52 

.52 

.52 

.52 

.52 

.52 

.52 

9.386438 

9.15 

9.15 

9.13 

9.12 

9.12 

9.10 

9.10 

9.08 

9.08 

9.05 

10.613562 

19 

42 

.374452 

.987465 

.386987 

.613013 

1 18  . 

43 

.374970 

.987434 

.387536 

.612464 

17 

44 

.375487 

.987403 

.388084 

.611916 

16 

45 

.376003 

.987372 

.388631 

.611369 

15 

46 

.376519 

.987341 

.389178 

.610822 

14 

47 

.377035 

.987310 

.389724 

.610276 

13 

48 

.377549 

.987279 

.390270 

.609730 

12 

49 

.378063 

.987248 

.390815 

.609185 

11 

50 

.378577 

.987217 

.391360 

.608640 

10 

51 

9-379089 

8.53 

8.53 

8.52 

8.50 

8.48 

8.48 

8.48 

8.45 

8.45 

9.987186 

.52 

.52* 

.53 

.52 

.52 

.53 

.52 

.52 

.53 

9.391903 

9.07 

9.03 

9.03 

9.03 

9.02 

9.00 

9.00 

8.98 

8.97 

10.608097 

9 

52 

.379601 

.987155 

.392447 

.607553 

8 

53 

.380113 

.987124 

.392989 

.607011 

7 

54 

.380624 

.987092 

.393531 

.606469 

6 

55 

.381134 

.987061 

.394073 

.605927 

5 

56 

.381643 

.987030 

.394614 

.605386 

4 

57 

.382152 

.986998 

.395154 

.604846 

3 

58 

.382661 

.986967 

.395694 

.604306 

2 

59 

.383168 

.986936 

.396233 

.603767 

1 

60 

9.383675 

9.986904 

9.396771 

10.603229 

0 

/ 

1 Cosine. 

D.  1".  j 

Sine. 

D.  1". 

Cotang. 

D.  l'\ 

| Tang. 

/ 

211 


103 


76* 


14 


TABLE  XTI.  LOGARITHMIC  SIKES. 


165° 


/ 

1 Sine. 

D.  1". 

Cosine. 

D.  1". 

Tang. 

U 

1 

Cotang, 

/ 

0 

9.383675 

8.45 

8.42 

8.42 

8.42 

8.40 

8.38 

8.38 

8.37 

8.35 

8.35 

8.33 

9.986904 

.52 

.53 

.53 

.52 

.53 

.53 

.52 

.53 

.53 

.53 

.53 

9.396771 

8.97 

8.95 

8.95 

8.93 

8.93 

8.92 

8.90 

8.90 

8.88 

8.88 

8.87 

10.603229 

60 

1 

.384182 

.986873 

.397309 

.602691 

59 

2 

.384687 

.986841 

.397846 

.602154 

58 

3 

.385192 

.986809 

.398383 

.601617 

57 

4 

.385697 

.986778 

.398919 

.601081 

56 

5 

.386201 

.986746 

.399455 

.600545 

55 

6 

.386704 

.986714 

.399990 

.600010 

54 

7 

.387207 

.986683 

.400524 

.599476 

53 

8 

.387709 

.986651 

.401058 

.598942 

52 

9 

.388210 

.986619 

.401591 

.598409 

51 

10 

.388711 

.986587 

.402124 

.597876 

50 

11 

9.389211 

8.33 

8.32 

8.30 

8.30 

8.28 

8.27 

8.27 

8.27 

8.23 

8.23 

9.986555 

.53 

.53 

.53 

.53 

.53 

.53 

.53 

.53 

.55 

.53 

9.402656 

8.85 

8.85 

8.85 

Q GO 

10.597344 

49 

12 

.389711 

.986523 

.403187 

.596813 

48 

13 

.390210 

.986491 

.403718 

.596282 

47 

14 

.390708 

.986459 

.404249 

.595751 

46 

15 

.391206 

.986427 

.404778 

O.O/O 

8.83 
8.80 
8.80 
8.80 
o 170 

.595222 

45 

16 

.391703 

.986395 

.405308 

.594692 

44 

17 

.392199 

.986363 

.405836 

.594164 

43 

18 

.392695 

.986331 

.406364 

.593636 

42 

19 

.393191 

.986299 

.406892 

.593108 

41 

20 

.393685 

.986266 

.407419 

8.77 

.592581 

40 

21 

9.394179 

8.23 

8.22 

8.20 

8.20 

8.18 

8.18 

8.15 

8.17 

8.15 

8.13 

9.986234 

.53 

.55 

.53 

.55 

.53 

.55 

.53 

.55 

.53 

.55 

9.407945 

8.77 

8.75 

8.75 

8.73 

8.73 

8.72 

Q 79 

10.592055 

39 

22 

.394673 

.986202 

.408471 

.591529 

38 

23 

.395166 

.986169 

.408996 

.591004 

37 

24 

.395658 

.986137 

.409521 

.590479 

36 

25 

.396150 

.986104 

.410045 

.589955 

35 

26 

.396641 

.986072 

.410569 

.589431 

34 

27 

.397132 

.986039 

.411092 

.588908 

33 

28 

.397621 

.986007 

.411615 

0 . i 6 

8.70 

8.68 

8.68 

.588385 

32 

29 

.398111 

.985974 

.412137 

.587863 

31 

30 

.398600 

.985942 

.412658 

.587342 

30 

31 

9.399088 

8.12 

8.12 

8.12 

8.10 

8.08 

8.08 

8.07 

8.05 

8.05 

8.05 

9.985909 

.55 

.55 

.53 

.55 

.55 

r;-. 

9.413179 

8.67 

8.67 

8.65 

8.65 

8.63 

8.63 

8.62 

8.60 

8.60 

8.60 

10.586821 

29 

32 

. 399575 

.985876 

.413699 

.586301 

28 

33 

.400062 

.985843 

.414219 

.585781 

27 

34 

.400549 

.985811 

.414738 

.585262 

26 

35 

.401035 

.985778 

.415257 

.584743 

25 

36 

.401520 

.985745 

.415775 

.584225 

24 

37 

.402005 

.985712 

. o > 
.55 
.55 
.55 
.55 

.416293 

.583707 

23 

38 

.402489 

.985679 

.416810 

.583190 

22 

39 

.402972 

.985646 

.417326 

.582674 

21 

40 

.403455 

.985613 

.417842 

.582158 

20 

41 

9.403938 

8.03 
8.02 
8.02 
8.00 
ty  oQ 

9.985580 

.55 

.55 

.57 

.55 

.55 

.55 

.57 

.55 

.57 

.55 

9.418358 

8.58 

8.57 

8.57 

8.57 

8.55 

8.55 

8.53 

8.52 

8.52 

8.50 

10.581642 

19 

42 

.404420 

.985547 

.418873 

.581127 

18 

43 

.404901 

.985514 

.419387 

.580613 

17 

44 

.405382 

.985480 

.419901 

.580099 

16 

45 

.405862 

.985447 

.420415 

.579585 

15 

43 

.406341 

7.98 

7.98 

7.97 

7.95 

7.95 

.985414 

.420927 

.579073  ! 

14 

47 

.406820 

.985381 

.421440 

.578560 

13 

48 

.407299 

.985347 

.421952 

.578048 

12 

49 

.407777 

.985314 

.422463 

.577537 

11 

50 

.408254 

.985280 

.422974 

.577026 

10 

51 

9.408731 

7.93 

7.93 

7.92 

7.92 

7.90 

7.88 

7.88 

7.87 

7.87 

9.985247 

.57 

.55 

.57 

.55 

.57 

.57 

.57 

.55 

.57 

9.423484 

8.48 

8.50 

8.47 

8.47 

8.47 

8.45 

8.45 

8.43 

8.42 

10.576516 

9 

52 

.409207 

.985213 

.423993 

.576007 

8 

53 

.409682 

.985180 

.424503 

.575497 

7 

54 

.410157 

.985146 

.425011 

.574989 

6 

55 

.410632 

.985113 

.425519 

.574481 

5 

56 

.411106 

.985079 

.426027 

.573973 

4 

57 

.411579 

.985045 

.426534 

.573466 

3 

58 

.412052 

.985011 

.427041 

.572959 

2 

59 

.412524 

.984978 

.427547 

.572453 

1 

60 

9.412996 

9.984944 

9.428052 

10.571948 

0 

' 

Cosine. 

D.  1". 

Sine.  | 

D.  1". 

Cotang. 

D.  1\  1 

Tang.  | 

/ 

104' 


212 


75 


15 


COSINES,  TANGENTS,  AND  COTANGENTS. 


1644 


' 

Sine. 

D.  1". 

Cosine. 

d.  r. 

Tang. 

d.  r. 

Cotang. 

' 

0 

1 

9.412996 

.413467 

7.85 

7.85 

7.83 

7.83 

9.984944 

.984910 

.57 

.57 

.57 

.57 

.57 

9-428052 

.428558 

8.43 

8.40 

8.40 

8.40 

8.38 

8.37 

8.37 

8.37 

8.35 

8.33 

8.33 

10.571948 
.571442  1 

60 

59 

2 

.413938 

.984876 

.429062 

. 57u938 

58 

3 

.414408 

.984842 

. 429506 

.570434 

57 

4 

.414878 

.981808 

.430070 

.569930 

56 

5 

.415347 

7.82 

7.80 

7.80 

.984774 

.430573 

.569427 

55 

C 

.415815 

.984740 

.5* 

.57 

.57 

.57 

.58 

.57 

.431075 

.568925 

54 

.416283 

.984706 

.431577 

.568423 

53 

8 

.416751 

7.80 

7.77 

7.78 
7.77 

.984672 

.432079 

.567921 

52 

9 

.417217 

.984638 

.432580 

.567420 

51 

10 

.417684 

.984603 

.433080 

.566920 

50 

11 

9.418150 

9.984569 

9.433580 

8.33 

8.32 

8.32 

8.30 

8.28 

8.28 

8.28 

8.27 

8.27 

8.25 

10.566420 

49 

12 

.418615 

7.75 

.984535 

.58 

.434080 

.565920 

48 

13 

.419079 

7.  <3 

.984500 

.434579 

.565421 

47 

14 

.419544 

7.75 

7.72 

.984466 

.435078 

.564922 

46 

15 

.420007 

.984432 

.58 

.435576 

.564424 

45 

16 

.420470 

7.72 

.984397 

.436073 

.563927 

44 

17 

.420933 

7.72 

7.70 

7.70 

7.68 

7.67 

.984363 

.58 

.57 

.58 

.58 

.436570 

.563430 

43 

18 

.421395 

.984328 

.437067 

.562933 

42 

19 

.421857 

.984294 

.437563 

.562437 

41 

20 

.422318 

.984259 

.438059 

.561941 

40 

21 

9.422778 

9.984224 

9.438554 

8.23 

8.25 

8.22 

8.22 

8.22 

8.20 

8.20 

8.18 

8.18 

8.18 

10.561446 

89 

22 

.423238 

7.67 

.984190 

.58 

.58 

.58 

.58 

.58 

.57 

.58 

.58 

.60 

.439048 

.560952 

88 

23 

.423697 

7.65 

.984155 

.439543 

.560457 

87 

24 

.424156 

7.65 

.984120 

.440036 

.559964 

36 

25 

.424615 

7.65 

.984085 

.440529 

.559471 

85 

26 

.425073 

7.63 

.984050 

.441022 

.558978 

34 

27 

.425530 

7.62 

.984015 

.441514 

.558486 

33 

28 

425987 

7.62 

.983981 

.442006 

.557994 

32 

29 

.426443 

7.60 

.983946 

.442497 

.557503 

31 

30 

.426899 

7.60 

7.58 

.983911 

.442988 

.557012 

30 

31 

32 

9.427354 

.427809 

7.58 

9.983875 

.983840 

.58 

.58 

.58 

.58 

.58 

.60 

.58 

.58 

.60 

.58 

9.443479 

.443968 

8.15 

8.17 

8.15 

8.13 

8.13 

8.13 

.8.12 

8.10 

8.10 

8.10 

10.556521 

.556032 

29 

28 

33 

.428263 

7.57 

.983805 

.444458 

.555542 

27 

34 

.428717 

7.57 

.983770 

.444947 

.555053 

26 

35 

.429170 

7.55 

.983735 

.445435 

.554565 

25 

36 

.429623 

7.55 

.983700 

.445923 

.554077 

24 

37 

.430075 

7.53 

.983664 

.446411 

.553589 

23 

38 

.430527 

7.53 

.983629 

.446898 

.553102 

22 

39 

.430978 

7.52 

.983594 

.447384 

.552616 

21 

40 

.431429 

7.52 

7.50 

.983558 

.447870 

.552130 

20 

41 

42 

9.431879 

.432329 

7.50 

7.48 

7.47 

7.48 
7.45 

9.983523 

.983487 

.60 

.58 

.60 

.58 

.60 

.60 

.60 

.58 

.60. 

.60 

9.448356 

.448841 

8.08 

8.08 

8.07 

8.07 

8.05 

8.05 

8.05 

8.03 

8.02 

8.02 

10.551644 

.551159 

19 

IS 

43 

.432778 

.983452 

.449326 

.550674 

17 

44 

.433226 * 

.983416 

.449810 

.550190 

16 

45 

.433675 

.983381 

.450294 

.549706 

15  1 

46 

.434122 

.983345 

.450777 

.549223 

14 

47 

.434569 

7.45 

7.45 

7.43 

7.43 

7.42 

.983309 

.451260 

.548740 

13  | 

48 

.435016 

.983273 

.451743 

.548257 

12 

49 

.435462 

.983238 

.452225 

547775 

11 

50 

.435908 

.983202 

.452706 

.547294 

10 

51 

52 

9.436353 

.436798 

7.42 

7.40 

9.983166 

.983130 

.60 

.60 

.60 

.60 

.60 

.60 

.60 

.60 

.60 

9.453187 

.453668 

8.02 
8.00 
8.00 
>?  Qft 

10.546813 

.546332 

9 

8 

53 

.437242 

.983094 

1 .454148 

.545852 

7 

54 

.437686 

7.40 

7.38 

7.38 

7.37 

7.37 

.983058 

1 .454628 

.545372 

6 

55 

.438129 

.983022 

.455107 

( . oO 

7.98 

7.97 

7.97 

7.95 

7.95 

. 544C93 

5 

56 

.433572 

.982986 

.455586 

.544414 

4 

57 

.439014 

.982950 

.456064 

.543936 

3 

58 

.439156 

.982914 

.456542 

.543458 

o 

59 

.439897 

7.35 

7.35 

.982878 

.457019 

.542981 

1 

60 

9.440338 

9.982842 

9.457496 

10.542504 

0 

/ 

I Cosine. 

D.  1". 

Sine. 

1 D.  v7\ 

i Cotang. 

D.  1". 

Tang. 

/ 

105°  74° 


213 


16°  TABLE  XTT.  LOGARITHMIC  SXMES,  163d 


/ 

Sine. 

D.  r. 

Cosine. 

D.  1". 

Tang. 

D.  1". 

Cotang. 

> 

0 

9.440338 

7.33 
7.33 
7.33 
7 30 

9.982842 

.62 

.60 

.60 

.62 

.60 

.60 

.62 

.60 

.62 

.62 

.60 

.62 

.62 

.60 

.62 

.62 

.62 

.62 

.62 

.62 

.62 

.62 
.62 
.62 
.62 
.63 
.62 
.62  ' 
.63 
.62 
.62 

9.457496 

7.95 

7.93 

7.93 

7.92 

7.92 

7.90 

7.90 

7.90 

7.88 

7.87 

7.88 

7.85 

7.87 

7.83 

7.85 

7.83 

7.83 

7.82 

7.82 

7.80 

7.80 

7.78 

i?Q 

10.542504 

60 

1 

.440778 

.982805 

.457973 

.542027 

59 

2 

.441218 

.982769 

.458449 

.541551 

58 

3 

.441658 

.982733 

.458925 

.541075 

57 

4 

.442096 

7.32 

7.30 

7.28 

7.28 

7.28 

7.27 

7.25 

.982696 

.459400 

.540600 

56 

5 

.442535 

.982660 

.459875 

.540125 

i 55 

6 

.442973 

.982624 

.460349 

.539651 

54 

7 

.443410 

.982587 

.460823 

.539177 

| 53 

8 

.443847 

.982551 

.461297 

.538703 

52 

9 

.444284 

.982514 

.461770 

.538230 

51 

10 

11 

.444720 

9.445155 

.982477 

9.982441 

.462242 

9.462715 

.537758 

10.537285 

, 50 
49 

12 

.445590 

7.25 

7.23 

7.23 

7.22 

7.22 

7.20 

7.20 

7.18 

7.18 

7.17 

7.17 

7.17 

7.15 

7.13 

7.13 

7.13 

7.12 

7.12 

7.10 

.982404 

.463186 

.536814 

48 

13 

.446025 

.982367 

.463658 

.536342 

1 47 

14 

.446459 

.982331 

.464128 

.535872 

46 

15 

.446893 

.982294 

.464599 

.535401 

45 

16 

.447326 

.982257 

.465069 

.534931 

44 

17 

.447759 

.982220 

.465539 

.534461 

43 

18 

.448191 

.982183 

.466008 

.533992 

42 

19 

.448623 

.982146 

.466477 

.533523 

41 

20 

21 

.449054 

9.449485 

.982109 

9.982072 

.466945 

9.467413 

.533055 

10.532587 

40 

39 

22 

.449915 

.982035 

.467880 

.532120 

38 

23 

.450345 

.981998 

. .468347 

*7 

.531653 

37 

24 

.450775 

.981961 

.468814 

< . <o 

•7  77 

.531186 

36 

25 

.451204 

.981924 

.469280 

< . < ( 
7.77 
7.75 

ty  vk 

.530720 

35 

26 

.451632 

.981886 

.469746 

.530254 

34 

27 

.452060 

.981849 

.470211 

.529789 

33 

28 

.452488 

.981812 

.470676 

( . 40 

yy  rn 

.529324 

32 

29 

.452915 

.981774 

.471141 

4.40 

.528859 

31 

30 

.453342 

.981737 

.471605 

4 . 4 o 

7.73 

.528395 

30 

31 

9.453768 

7.10 

7.08 

7.08 

7.08 

7.07 

7.05 

7.05 

7.05 

7.03 

7.03 

7.02 

7.02 

7.00 

7.00 

7.00 

6.98 

6.98 

6.97 

6.97 

6.95 

9.981700 

.63 

.62 

.63 

.63 

.62 

.63 

.63 

.62 

.63 

.63 

.63 

.63 

.63 

.63 

.63 

.63 

.63 

.63 

.63 

.65 

'9.472069 

7 79 

10.527931 

29 

32 

.454194 

.981662 

.472532 

t . 

7 79 

.527468 

28 

33 

.454619 

.981625 

.472995 

7 70 

.527005 

27 

34 

.455044 

.981587 

.473457 

1 . IU 

7 70 

.526543 

26 

35 

.455469 

.981549 

.473919 

t . 

7 70 

.526081 

25 

36 

.455893 

.981512 

.474381 

7. '68 
7.68 

7 07 

.525619 

24 

37 

.456316 

.981474 

.474842 

.525158 

23 

38 

.456739 

.981436 

.475303 

.524697 

22 

39 

.457162 

.981399 

.475763 

1 . Ol 

7.67 

7.67 

7.65 
7.65 
7.63 
7.63 
7.63 
7.62 
7.62 
7.60 
. 7.60 
7.60 

.524237 

21 

40 

41 

.457584 

9.458006 

.981361 

9.981323 

.476223 

9.476683 

.523777 

10.523317 

20 

19 

42 

.458427 

.981285 

.477142 

.522858 

18 

43 

.458848 

.981247 

.477601 

.522399 

17 

44 

.459268 

.981209 

.478059 

.521941 

16 

45 

.459688 

.981171 

.478517 

.521483  1 

15 

46 

.460108 

.981133 

.478975 

.521025  1 

14 

47 

.460527 

.981095 

.479432 

.520568 

13 

48 

.460946 

.981057 

.479889 

.520111 

12 

49 

.461364 

.981019 

.480345 

.519655 

11 

50 

.461782 

.980981 

.480801 

.519199 

10 

51 

9.462199 

6.95 

6.93 

6.93 

6.93 

6.92 

6.92 

6.90 

6.90 

6.88 

9.980942 

.63 

.63 

.65 

.63 

.65 

.63 

.65 

.63 

.65 

9.481257 

7.58 

7.58 

10.518743 

9 

52 

.462616 

.980904 

! .481712 

.518288 

8 

53 

.463032 

.980866 

.482167 

.517833 

7 

54 

.463448 

.980827 

.482621 

4.0  4 

7.57 

7 *»7 

.517379 

6 

55 

.463864 

. 9807'89 

.483075 

.516925 

5 

56 

.464279 

. 9807'50 

.483529 

i .0  ( 

7.55 

7.55 

7.53 

7.53 

.516473 

4 

57 

.464694 

.980712 

.483982 

.51601® 

3 

58 

.465108 

.980673 

.484435 

.515565 

2 

59 

.465522 

.980635 

.484887 

.516118 

1 

60 

9.465935 

9.980596 

9.485339 

10.53 3663 

0 

Cosine. 

TxrT 

Sine. 

D.  1". 

Cotang. 

D.  r. 

Tang.  | 

/ 

100° 


214 


73' 


COSINES,  TANGENTS,  ANI)  COTANGENTS. 


162* 


1*0 


/ 

Sine. 

V-  r. 

Cosine. 

D.  1". 

Tang. 

D.  1". 

Cotang. 

/ 

0 

9.465935 

6.88 

6.88 

6.87 

6.87- 

6.85 

6.85 

6.83 

6.83 

6.83 

6.82 

6.82 

9.980596 

.63 

.65 

.65 

.63 

.65 

.65 

.65 

.65 

.65 

.65 

.65 

.65 

.65 

.65 

.67 

.65 

.65 

.65 

.67 

.65 

.67 

.65 

.67 

.65 

.67 

.65 

9.485339 

7.53 

7.52 

7.52 

7.50 

7.50 

7.50 

7.48 

7.48 

7.48 

7 /17 

10.514661 

60 

1 

.466348 

.980558 

.485791 

.514209 

59 

g 

.466761 

.980519 

.486242 

.513758 

58 

3 

.467173 

.980480 

.486693 

.513307  1 

57 

4 

.467585 

.980442 

.487143 

.512857 

56 

5 

.467996 

.980403 

.487593 

.512407 

55 

6 

.465407 

.980364 

.488043 

.511957 

54 

.468817 

.980325 

.488492 

.511508 

53 

8 

.469227 

.980286 

.488941 

.511059 

52 

9 

.469637 

.980247 

.489390 

.510610 

51 

10 

11 

.470046 

9.470455 

.980208 

9.980169 

.489838 

9.490286 

( .4* 

7.47 

7.45 

7.45 

7.45 

7.43 

7.43 

7.43 

7.42 

7.40 

7.42 

7.40 

7.38 
7.40 
7 38 

.510162 
10.509714  1 

50 

49 

12 

.470863 

6.80 

6.80 

6.80 

.980130 

.490733 

.509267 

48 

13 

.471271 

.980091 

.491180 

.508820 

47 

14 

.471679 

.980052 

.491627 

.508373 

46 

15 

.472086 

6.  <o 

6.77 

6.77 

6.77 

6.77 

6.75 

6.73 

6.73 

6.73 

6.72 

6.72 

.980012 

.492073 

.507927 

45 

16 

.472492 

.979973 

.492519 

.507481 

44 

17 

.472898 

.979934 

.492965 

.507035 

43 

18 

.473304 

.979895 

.493410 

.506590 

42 

19 

.473710 

.979855 

.493854 

.506146 

41 

20 

21 

.474115 

9.474519 

.979816 

9.979776 

.494299 

9.494743 

.505701 

10.505257 

40 

39 

22 

.474923 

.979737 

.495186 

.504814 

38 

23 

.475327 

.979697 

.495630 

.504370 

37 

24 

.475730 

.979658 

.496073 

7.37 

7.37 

7.37 

7.37 

7.35 

7.33 

7.35 

7.33 

7.32 

7.32 

7.32 

7.32 

7.30 

7.30 

7.28 

7.28 

7.28 

.503927 

! 36 

25 

.476133 

.979618 

.496515 

.503485 

1 35 

26 

.476536 

6.72 

6.70 

6.70 

6.68 

6.68 

6.67 

6.67 

6.67 

6.65 

6.65 

6.65 

6.63 

6.62 

6.62 

6.62 

6.62 

. 979579 

.496957 

.503043 

• 34 

27 

.476938 

.979539 

.67 

.497399 

.502601 

33 

28 

.477340 

.979499 

.497841 

.502159 

32 

29 

.477741 

.979459 

.67 

.65 

.67 

.67 

.67 

.67 

.498282 

.501718 

31 

30 

31 

.478142 

9.478542 

.979420 

9.979380 

.498722 

9.499163 

.501278 

10.500837 

30 

29 

32 

.478942 

.979340 

.499603 

.500397 

28 

33 

.479342 

.979300 

.500042 

.499958 

i 27 

34 

.479741 

.979260 

.500481 

.499519 

26 

35 

.480140 

.979220 

.67 

.67 

.67 

.67 

.68 

.67 

.67 

.500920 

.499080 

25 

36 

.480539 

.979180 

.501359 

.498641 

24 

37 

.480937 

.979140 

.501797 

.498203 

23 

38 

.481334 

.979100 

.502235 

.497765 

22 

39 

.481731 

. 97'9059 

.502672 

.497328 

21 

40 

.482128 

.979019 

.503109 

.496891 

20 

41 

9.482525 

6.60 

6.58 

6.60 

'6.58 

6.57 

6.57 

6.57 

6.55 

6.55 

6.53 

6.55 

6.52 

6.53 
6.52 
6.50 
6.50 
6.50 
6.48 
6.48 

9.978979 

.67 

.68 

.67 

.68 

.67 

9.503546 

7.27 

7.27 

7.27 

7.25 

7.25 

7.25 

7.23 

7.23 

7.23 

7.22 

7.22 

7.22 

7.20 

7.18 

7.20 

7.18 

7.18 

7.17 

7.17 

10.496451 

! 19 

42 

.482921 

.978939 

.503982 

.496018 

18 

43 

.483316 

.978898 

.504418 

.495582 

17 

41 

.483712 

.978858 

.504854 

.495146 

16 

45 

.484107 

.978817 

.505289 

.494711 

15 

46 

.484501 

. 978777 

.505724 

.494276 

14 

47 

.484895 

.978737 

.67 

.68 

.68 

.506159 

.493841 

13 

48 

.485289 

.978696 

. 506593 

.493407 

12 

49 

.485682 

.978655 

.507027 

.492973 

11 

50 

51 

.486075 

9.486467 

.978615 

9.978574 

.68 

.68 

.507460 

9.507893 

.492540 

10.492107 

10 

9 

52 

.486860 

.978533 

.508326 

.491674 

8 

53 

.487251 

.978493 

.0* 

.68 

.68 

.68 

.68 

.68 

.68 

.68 

. 508759 

.491241 

7 

54 

.487643 

.978452 

.509191 

.490309 

6 

55 

.488034 

.978411 

.509622 

.490378 

5 

56 

.488424 

.978370 

.510054 

.489946 

4 

57 

.488814 

978329 

! .510485 

.489515 

3 

58 

.489204 

.978288 

S .510916 

.489084 

2 

59 

.489593 

.978247 

.511346 

.488654 

1 

60 

9.489982 

9.978206 

. 9.511776 

10.488224 

0 

' 

Cosine. 

i D V . 

1 Sine. 

| D.  1". 

Cotang. 

1 D.  1\ 

1 Tang. 

/ 

107' 


215 


72 


18°  table  xii.  Logarithmic  sines. 


' 

Sine. 

D.  1". 

Cosine. 

D.  1". 

Tang. 

D.  1". 

i Cotang. 

/ 

0 

1 

2 

3 

4 

5 
C 

7 

8 
9 

10 

11 

12 

13 

14 

15 

16 
17 
38 

19 

20 

21 

22 

23 

24 

25 

26 

27 

28 

29 

30 

31 

32 

33 

34 

35 

36 

37 

38 

39 

40 

41 

42 

43 

44 

45 

46 

47 

48 

49 

50 

51 

52 

53 

54 

55 

56 

57 

58 

59 

60 

9.489982 

.490371 

.490759 

.491147 

.491535 

.491922 

.492308 

.492695 

.493081 

.493466 

.493851 

9.494236 

.494621 

.495005 

.495388 

.495772 

.496154 

.496537 

.496919 

.497301 

.497682 

9.498064 

.498444 

.498825 

.499204 

.499584 

.499963 

.500342 

.500721 

.501099 

.501476 

9.501854 

.502231 

.502607 

.502984 

.503360 

.503735 

.504110 

.504485 

.504860 

.505234 

9.505608 

.505981 

.506354 

.506727 

.507099 

.507471 

.507843 

.508214 

.508585 

.508956 

9.509326 

.509696 

.510065 

.510434 

.510803 

.511172 

.511540 

.511907 

.512275 

9.512642 

6.48 

6.47 

6.47 

6.47 

6.45 

6.43 

6.45 

6.43 

6.42 

6.42 

6.42 

6.42 

6.40 

6.38 

6.40 

6.37 

6.38 
6.37 
6.37 
6.35 
6.35 

6.33 

6.35 

6.32 

6.33 
6.32 
6.32 
6.32 
6.30 
6.28 
6.30 

6.28 

6.27 

6.28 
6.27 
6.25 
6.25 
6.25 
6.25 
6.23 
6.23 

6.22 

6.22 

6.22 

6.20 

6.20 

6.20 

6.18 

6.18 

6.18 

6.17 

6.17 
6.15 
6.15 
6.15 
6.15 
6.13 
6.12 
6.13 
6.12 
- 

9.978206 

.978165 

.978124 

.978083 

.978042 

.978001 

.977959 

.977918 

.977877 

.977835 

.977794 

9.977752 
.977711 
.977669 
.977628 
. 977586 
.977544 
.977503 
.977461 
.977419 
.977377 

9.977335 

.977293 

.977251 

.977209 

.977167 

.977125 

.977083 

.977041 

.976999 

.976957 

9.976914 

.976872 

.976830 

.976787 

.976745 

.976702 

.976660 

.976617 

.976574 

.976532 

9.976489 

.976446 

.976404 

.976361 

.976318 

.976275 

.976232 

.976189 

.976146 

.976103 

9.976060 

.976017 

.975974 

.975930 

.975887 

.97*5844 

.975800 

.975757 

.975714 

9.975670 

.68 

.68 

.68 

.68 

.68 

.70 

.68 

.68 

.70 

.68 

.70 

.68 
.7*0 
.68 
.70 
.70 
.68 
.7*0 
.70 
.70 
. .70 

.70 

.70 

.70 

.70 

.70 

.70 

.70 

.70 

.70 

.7*2 

.70 

.70 

.7*2 

.70 

.72 

.70 

.7*2 

.72 

.70 

.7*2 

.72 

.70 

.7*2 

.72 

.7*2 

.72 

^72 

.72 

.72 

.7*2 

.72  • 
• .7*2 

.73 
.7*2 
.72 
.73 
.72 
.72 
.73 

9.511776 

.512206 

.512635 

.513064 

.513493 

.513921 

.514349 

.514777 

.515204 

.515631 

.516057 

9.516484 

.516910 

.517335 

.517761 

.518186 

.518610 

.519034 

.519458 

.519882 

.520305 

9.520728 

.521151 

.521573 

.521995 

.522417 

.522838 

.523259 

.523680 

.524100 

.524520 

9.524940 

.525359 

.525778 

.526197 

.526615 

.527033 

.527*451 

.527868 

.528285 

.528702 

9.529119 

.529535 

.529951 

.530366 

.530781 

.531196 

.531611 

.532025 

.532439 

.532853 

9.533266 

.533679 

.534092 

.534504 

.534916 

.535328 

.5357*39 

.536150 

.536561 

9.536972 

7.17 

7.15 

7.15 

7.15 

7.13 

7.13 

7.13 

7.12 

7.12 

7.10 

7.12 

7.10 

7.08 

7.10 

7.08 

7.07 

7.07 

7.07 

7.07 

7.05 

7.05 

7.05 

7.03 

7.03 

7.03 

7.02 

7.02 

7.02 

7.00 

7.00 

7.00 

6.98 

6.98 

6.98 

6.97 

6.97 

6.97 

6.95 

6.95 

6.95 

6.95 

6.93 

6.93 

6.92 

6.92 

6.92 

6.92 

6.90 

6.90 

6.90 

6.88 

6.88 

6.88 

6.87 

6.87 

6.87 

6.85 

6.85 

6.85 

6.85 

10.488224 
.487794 
.487365 
.486936 
.486507 
.486079 
.485651 
.485223 
.484796 
.484369 
.483943 
10.483516 
.483090 
.482665 
.482239 
.481814 
.481390 
.480966 
.480542 
.480118 
.479695 
10.479272 
.478849 
* .478427 
.478005 
.477583 
.477162 
.476741 
.476320 
.475900 
.475480 

10.475060 

.474641 

.474222 

.473803 

.473,385 

.472967 

.472549 

.472132 

.471715 

.471298 

10.47*0881 
.47*0465 
.470049 
„ .469634 
.469219 
.468804 
.468389 
.467*97*5 
.467561 
.467147 

10.466734 

.466321 

.465908 

.465496 

.465084 

.46467*2 

.464261 

.463850 

.468489 

10.468038 

60 

59 

58 

57 

56 

55 

54 

53 

52 

51 

50 

49 

48 

47 

46 

45 

44 

43 

42 

41 

40 

39 

38 

37 

36 

35 

34 

33 

32 

31 

30 

29 

28 

27 

26 

25 

24 

23 

22 

21 

20 

19 

18 

17 

16 

15 

14 

13 

12 

11 

10 

9 

8 

7 

6 

5 

4 

3 

2 

1 

0 

' 1 Cosine. 

D.  1".  1 

Sine.  | 

I).  1". 

Cotang.  1 

D.  1\  1 

Tang. 

COSINES,  TANGENTS,  AND  COTANGENTS. 


160* 


l9° 


' Sine. 

D.  1". 

Cosine. 

D.  V. 

Tang. 

D.  1". 

Cotang. 

/ 

0 1 

1 ! 

2 1 

t! 

I 

7 

S 

9 

10 

II 
12 

13 

14 

15 

16 

17 

18 

19 

20 

21 

22 

23 

24 

25 

26 

27 

28 

29 

30 

31 

32 

33 

34 

35 

36 

37 

38 

39 

40 

41 

42 

43 

44 

45 

46 

47 

48 

49 

50 

51 

52 

53 

54 

55 
• 56 
i 57 

58 

59 

60 

9.512642 

.513009 

.513375 

.513741 

.514107 

.514472 

.514837 

.515202 

.515566 

.515930 

.516294 

9.516657 

.517020 

.517382 

.517745 

.518107 

.518468 

.518829 

.519190 

.519551 

.519911 

9.520271 

.520631 

.520990 

.521349 

.521707 

.522066 

.522424 

.522781 

.523138 

.523495 

9.523852 

.524208 

.524564 

.524920 

.525275 

.525630 

.525984 

.526339 

.526693 

.527046 

9.527400 

.527753 

.528105 

.528458 

.528810 

.529161 

.529513 

.529864 

.530215 

.530565 

9.530915 

.531265 

.531614 

.531963 

.532312 

.532661 

.533009 

.533357 

.533704 

9.534052 

6.12 

6.10 

6.10 

6.10 

6.08 

6.08 

6.08 

6.07 

6.07 

6.07 

6.05 

6.05 

6.03 

6.05 

6.03 

6.02 

6.02 

6.02 

6.02 

6.00 

6.00 

6.00 

5.98 

5.98 

5.97 

5.98 
5.97 
5.95 
5.95 
5.95 
5.95 

5.93 

5.93 

5.93 

5.92 

5.92 

5.90 

5.92 

5.90 

5.88 

5.90 

5.88 

5.87 

'5.88 

5.87 

5.85 

5.87 

5.85 

5.85 

5.83 

5.83 

5.83 

5.82 

5.82 

5.82 

5.82 

5.80 

5.80 

5.78 

5.80 

9.975670 
.975627 
.975583 
.975539 
.975496 
.975452 
. 975408 
.975365 
.975321 
.975277 
.975233 

9.975189 

.975145 

.975101 

.975057 

.975013 

.974969 

.974925 

.974880 

.974836 

.974792 

9.974748 

.974703 

.974659 

.974614 

.974570 

.974525 

.974481 

.974436 

.974391 

.974347 

9.974302 

.974257 

.974212 

.974167 

.974122 

.974077 

.974032 

.973987 

.973942 

.973897 

9.973852 

.973807 

.973761 

.973716 

.973671 

.973625 

.973580 

.973535 

.973489 

.973444 

9.973398 

.973352 

.973307 

.973261 

.973215 

.973169 

.973124 

.973078 

.973032 

9.972986 

.72 

.73 

A3 

.72 

.73 

.73 

.72 

.73 

.73 

.73 

.73 

.73 

.73 

.73 

.73 

.73 

.73 

.75 

.73 

.73 

.73 

.75 

.73 

.75 

.73 

.75 

.73 

.75 

.75 

.73 

.75 

.75 

.75 

.75 

.75 

.75 

.75 

.75 

.75 

.75 

.75 

.75 

.77 

.75 

.75 

.75 

.75 

.77 

.75 

..77 

.75 

.77 

.77 

.77 

.75 

.77 

.77 

.77 

9.536972 

.537382 

.537792 

.538202 

.538611 

.539020 

.539429 

.539837 

.540245 

.540653 

.541061 

9.541468 

.541875 

.542281 

.542688 

.543094 

.543499 

.543905 

.544310 

.544715 

.545119 

9.545524 

.545928 

.546331 

.546735 

.547138 

.547540 

.547943 

.548345 

.548747 

.549149 

9.549550 

.549951 

.550352 

.550752 

.551153 

.551552 

.551952 

.552351 

.552750 

.553149 

9.553548 

.553946 

.554344 

.554741 

.555139 

.555536 

.555933 

.556329 

.556725 

.557121 

9.557517 

.557913 

.558308 

.558703 

.559097 

.559491 

.559885 

.560279 

.560673 

9.561066 

6.83 

6.83 

6.83 

6.82 

6.82 

6.82 

6.80 

6.80 

6.80 

6.80 

6.78 

6.78 

6.77 

6.78 
6.77 
6.75 
6.77 
6.75 
6.75 
6.73 
6.75 

6.73 

6.72 

6.73 
6.72 
6.70 
6.72 
6.70 
6.70 
6.70 
6.68 

6.68 

6.68 

6.67 

6.68 
6.65 
6.67 
6.65 
6.65 
6.65 
6.65 

6.63 

6.63 

6.62 

6.63 

6.62 

6.62 

6.60 

6.60 

6.60 

6.60 

6.60 

6.58 

6.58 

6.57 

6.57 

6.57 

6.57 

6.57 

6.55 

10.463028 

.462618 

.462208 

.461798 

.461389 

.460980 

.460571 

.460163 

.459755 

.459347 

.458939 

10.458532 

.458125 

.457719 

.457312 

.456906 

.456501 

.456095 

.455690 

.455285 

.454881 

10.454476 

.454072 

.453669 

.453265 

.452862 

.452460 

.452057 

.451655 

.451253 

.450851 

10.450450 

.450049 

.449648 

.449248 

.448847 

.448448 

.448048 

.447649 

.447250 

.446851 

.10.446452 

.446054 

.445656 

.445259 

.444861 

.444464 

.444067 

.443671 

.443275 

.442879 

10.442483 

.442087 

.441692 

.441297 

.440903 

.440509 

.440115 

.439721 

.439327 

10.438934 

60 

59 

58 

57 

56 

55 

54 

53 

52 

51 

50 

49 

48 

47 

46 

45 

44 

43 

42 

41 

40 

39 

38 

37 

36 

35 

34 

33 

32 

31 

30 

29 
28 
27 
26 
25 
24 
23 
j 22 
21 
; 20 

19 
18 
17 
16 
! 15 
14 
! 13 
12 
11 
10 

9 

8 

6 

5 

4 

3 

2 

1 

0 

/ 

Cosine. 

1 d.  r. 

Sine. 

i d.  r. 

Cotang. 

D.  1". 

1 Tang. 

) - ' 

109' 


217 


70' 


20c 


TABLE  Xif.  LOGARITHMIC  SINES. 


159^ 


' 

Sine. 

D.  1". 

Cosine. 

D.  1". 

Tang. 

D.  1". 

Cotang. 

' 

9 

2 

3 

4 

5 

6 

8 

9 

10 

11 

12 

13 

14 

15 

16 

17 

18 

19 

20 

21 

22 

23 

24 

25 

26 

27 

28 

29 

30 

31 

32 

33 

34 

35 

36 

37 

38 

39 

40 

41 

42 

43 

44 

45 

46 

47 

48 

49 

50 

51 

52 

53 

54 

55 

56 

57 

58 

59 

60 

£.534052 

.534399 

.534745 

.535092 

.535438 

.535783 

.536129 

.536474 

.536818 

.537163 

.537507 

9.537851 

.538194 

.538538 

.538880 

.539223 

.539565 

.539907 

.540249 

.540590 

.540931 

9.541272 

.541613 

.541953 

.542293 

.542632 

.542971 

.543310 

.543649 

.543987 

.544325 

9.544663 

.545000 

.545338 

.545674 

.546011 

.546347 

.546683 

.547019 

.547354 

.547689 

9.548024 

.548359 

.548693 

.549027 

.549360 

.549693 

.550026 

.550359 

.550692 

.551024 

9.551356 

.551687 

.552018 

.552349 

.552680 

.553010 

.553341 

.553670 

.554000 

9.554329 

5.78 

5.77 

5.78 
5.77 
5.75 
5.77 
5.75 
5.73 
5.75 
5.73 
5.73 

5.72 

5.73 
5.70 
5.72 
5.70 
5.70 
5.68 
5.68 
5.68 
5.68 

5.68 

5.67 

5.67 

5.65 

5.65 

5.65 

5.65 

5.63 

5.63 

5.63 

5.62 

5.63 
5.60 
5.62 
5.60 
5.60 
5.60 
5.58 
5.58 
5.58 

5.58 
• 5.57 
5.57 
5.55 
5.55 
5.55 
5.55 
5.55 
5.53 
5.53 

5.52 

5.52 

5.52 

5.52 

5.50 

5.52 

5.48 

5.50 

5.48 

9.972986 

.972940 

.972894 

.972848 

.972802 

.972755 

.972709 

.972663 

.972617 

.972570 

.972524 

9.972478 
.972431 
.972385 
.972338 
.972291 
.972245 
.972198 
.972151 
.972105 
. .972058 

9.972011 

.971964 

.971917 

.971870 

.971823 

.971776 

.971729 

.971682 

.971635 

.971588 

9.971540 

.971493 

.971446 

.971398 

.971351 

.971303 

.971256 

.971208 

.971161 

.971113 

9.971066 
.971018 
.970970 
.970922 
.970874 
.970827 
. .970779 
.970731 
.970683 
.970635 

9.970586 

.970538 

.970490 

.970442 

.970394 

.970345 

.970297 

.970249 

.970200 

9.970152 

.77 

.77 

.77 

.77 

.78 

.77 

.77 

.77 

.78 

.77 

.77 

.78 

.77 

.78 

.78 

.77 

.78 

.78 

.77 

.78 

.78 

.78 

.78 

.78 

.78 

.78 

.78 

.78 

.78 

.78 

.80 

.78 

.78 

.80 

.78 

.80 

.78 

.80 

.78 

.80 

.78 

.80 

.80 

.80 

.80 

.78 

.80 

.80 

.80 

.80 

.82 

.80 

.80 

.80 

.80 

.82 

.80 

.80 

.82 

.80 

9.561066 
.561459 
! .561851 

! .562244 

j .562636 
.563028 
.563419 
.563811 
I .564202 
.564593 
.564983 

9.565373 

.565763 

.566153 

.566542 

.566932 

.567320 

.567709 

.568098 

.568486 

.568873 

9.569261 
I .569648 
! .570035 

| .570422 

I .570809 
.571195 
.571581 
.571967 
! .572352 

.572738 

9.573123 

.573507 

.573892 

.574276 

.574660 

.575044 

.575427 

.575810 

.576193 

.576576 

9.576959 
.577341 
.577723 
.578104 
.578486 
.578867 
.579248 
i .579629 
.580009 
.580389 

9.580769 
: .581149 

.581528 
.581907 
.582286 
.582665 
.583044 
.583422 
.583800 
9.584177 

6.55 

6.53 

6.55 

6.53 

6.53 

6.52 

6.53 
6.52 
6.52 
6.50 
6.50 

6.50 

6.50 

6.48 

6.50 

6.47 

6.48 
6.48 
6.47 
6.45 
6.47 

6.45 

6.45 

6.45 

6.45 

6.43 

6.43 

6.43 

6.42 

6.43 
6.42 

6.40 

6.42 

6.40 

6.40 

6.40 

6.38 

6.38 

6.38 

6.38 

6.38 

6.37 

6.37 

6.35 

6.37 

6.35 

6.35 

6.35 

6.33 

6.33 

6.33 

6.33 

6.32 

6.32 

6.32 

6.32 

6.32 

6.30 

6.30 

6.28 

10.438934 

.438541 

.438149 

.437756 

.437364 

.436972 

.436581 

.43618& 

.435798 

.435407 

.435017 

10.434627 

.434237 

.433847 

.433458 

.433068 

.432680 

.432291 

.431902 

.431514 

.431127 

10.430739 
. 430352 
.429965 
.42957'8 
.429191 
.428805 
.428419 
.428033 
.427648 
.427262 

10.426877 

.426493 

.426108 

.425724 

.425340 

.424956 

.424573 

.424190 

.423807 

.423424 

10.423041 

.422659 

.422277 

.421896 

.421514 

.421133 

.420752 

.420371 

.419991 

.419611 

10.419231 
.418851 
.418472 
.418093 
.417714 
.417335 
.416956 
.416578 
.41  (>200 

10.415823 

60 

59 

58 

57 

56 

55 

54 

53 

52 

51 

50 

49 

48 

47 

46 

45 

44 

43 

42 

41 

40 

39 

38 

37 

36 

35 

34 

33 

32 

31 

30 

29 

28 

27 

26 

25 

24 

23 

22 

21 

20 

19 

18 

17 

16 

15 

14 

13 

12 

11 

10 

9 

8 

7 

6 

5 

4 

3 

2 

1 

0 

' 

Cosine. 

d.  r.  11 

Sine. 

D.  1".  l| 

Cotang. 

D.  1". 

Tang.  | ' 

no* 


218 


69' 


COSINES,  TANGENTS,  AND  COTANGENTS* 

21* 


/ 

Sine. 

D.  1". 

Cosine. 

D.  1*. 

Tang. 

D.  r. 

Cotang. 

0 

9.554329 

5.48 

5.48 

5.47 

5.47 

5.47 

5.47 

5.45 

5.45 

5.45 

5.43 

5.43 

5.43 

5.42 

5.43 
5.42 
5.40 
5.42 
5.40 
5.40 
5.40 
5.38 

5.38 

5.38 

5.37 

5.37 

5.37 

5.37 

5.35 

5.37 

5.33 

5.35 

K QQ 

9.970152 

.82 

9.584177 

6.30 

6.28 

6.28 

6.28 

6.27 

6.28 
6.27 

10.415823 

1 

.554658 

.970103 

.584555 

.415415 

2 

.554987 

.970055 

.80 

.82 

.82 

.584932 

.415068 

3 

.555315 

.970006 

.585309 

.414691 

4 

.555643 

.969957 

.585686 

.414314 

5 

.555971 

.969909 

.80 

.82 

.82 

.82 

.80 

.82 

.82 

.82 

.82 

.586062 

.413938 

6 

.556299 

.969860 

.'586439 

.413561 

7 

.556626 

.969811 

.586815 

.413185 

8 

.556953 

.969762 

.587190 

6.25 

6.27 

6.25 

6.25 

6.25 

6.25 

6.23 

6.23 

6.23 

6.23 

6.22 

6.22 

6.22 

6.22 

6.20 

6.22 

6.20 

6.18 

6.20 

6.18 

6.18 

6.18 

6.18 

6.17 

6.17 

6.17 

6.17 

6.15 

6.15 

6.15 

6.15 

6.13 

6.15 

6.13 

6.13 

6.12 

6.13 

6.12 

6.12 

6.12 

6.10 

6.10 

6.10 

6.10 

6.10 

6.08 

6.08 

6.08 

6.08 

6.07 

6.08 
6.07 
6.07 

.412810 

9 

.557280 

.969714 

.587566 

.412434 

10 

11 

.557606 

9.557932 

.969665 

9.969616 

.587941 

9.588316 

.412059 

10.411684 

12 

.558258 

.969567 

.588691 

.411309 

13 

.558583 

.969518 

.589066 

.410934 

14 

.558909 

.969469 

.82 

.82 

.589440 

.410560 

15 

.559234 

.969420 

.589814 

.410186 

16 

.559558 

.969370 

.83 

.82 

.82 

.82 

.590188 

.409812 

17 

.559883 

.969321 

.590562 

.409438 

18 

.560207 

.969272 

.590935 

.409065 

19 

.560531 

.969223 

.591308 

.408692 

20 

21 

.560855 

9.561178 

.969173 

9.969124 

.83 

.82 

.82 

.83 

.82 

.83 

.82 

.83 

.591681 

9.592054 

.408319 

10.407946 

22 

.561501 

.969075 

.592426 

.407574 

23 

.561824 

.969025 

.592799 

.407201 

24 

.562146 

.968976 

.593171 

.406829 

25 

.562468 

.968926 

.593542 

.406458 

26 

.562790 

.968877 

.593914 

.406086 

27 

.563112 

.968827 

.594285 

.405715 

28 

.563433  ' 

.968777 

.83 

.82 

.594656 

.405344 

29 

.563755 

.968728 

.595027 

.404973 

30 

31 

.564075 

9.564396 

.968678 

9.968628 

.83 

.83 

.83 

.83 

.82 

.83 

.83 

.595398 

9.595768 

.404602 

10.404232 

32 

.564716 

D . OO 

5.33 

5.33 

5.33 

5.32 

5.32 

5.30 

5.32 

5.30 

5.30 

5.28 

5.30 

5.28 

5.28 

5.27 

5.27 

5.27 

5.27 

5.25 

5.27 

5.25 

5.23 

5.25 

5.23 

5.23 

5.22 

5.23 
5.22 
5.20 

.968578 

.596138 

.403862 

33 

.565036 

.968528 

.596508 

.403492 

34 

.565356 

.968479 

. 59687'8 

.403122 

35 

.565676 

.968429 

.597247 

.402753 

36 

.565995 

.968379 

.597616 

.402384 

37 

.566314 

.968329 

.83 

.85 

.83 

.83 

.83 

QQ 

.597985 

.402015 

38 

.566632 

.968278 

.598354 

.401646 

39 

.566951 

.968228 

.598722 

• .401278 

40 

41 

.567269 

9.567587 

.968178 

9.968128 

.599091 

9.599459 

.400909 

10.400541 

42 

.567904 

.968078 

.OO 

.85 

.83 

.599827 

.400173 

43 

.568222 

.968027 

.600194 

.399806 

44 

.568539 

.967977 

.600562 

.399438 

45 

.568856 

.967927 

.83 

.85 

.600929 

.399071 

46 

.569172 

.967876 

.601296 

.398704 

47 

.569488 

.967826 

.83 

.85 

.83 

.85 

.83 

.85 

.85 

.85 

.83 

.85 

.85 

.85 

.85 

.85 

.601663 

.398337 

48 

.569804 

.967775 

.602029 

.397971 

49 

.570120 

.967725 

.602395 

.397605 

50 

51 

.570435 

9.570751 

.967674 

9.967624 

• .602761 
9.603127 

.397239 

10.396873 

52 

.571066 

.967573 

.603493 

.3965,7 

53 

.571380 

.967522 

.603858 

.396142 

54 

.571695 

.967471 

.604223 

.395777 

55 

.572009 

.967421 

.604588 

.395412 

56 

.572323 

.967370 

.604953 

.395047 

57 

.572636 

.967319 

.605317 

.394683 

58 

.572950 

.967268 

.605682 

.394318 

59 

.573263 

.967217 

.606046 

.393954 

60 

9.573575 

9.967166 

9.606410 

10.393590 

* 7 

Cosine. 

D.  r. 

i Sine. 

D.  1\  1 

1 Cotang. 

D„  1". 

l Tang. 

158° 

/ 

60 

59 

58 

57 

56 

55 

54 

53 

52 

51 

50 

49 

48 

47 

46 

45 

44 

43 

42 

41 

40 

39 

38 

37 

36 

35 

34 

33 

32 

31 

30 

29 

28 

27 

26 

25 

24 

23 

22 

21 

20 

19 

18 

17 

16 

15 

14 

I 13 

| 12 

11 

1C 

9 

i 8 

7 

6 

I 5 

I 4 

i 3 

I 2 

] 1 

| 0 


nr 


819 


68' 


22* 


TABLE  XII.  LOGARITHMIC  SIKES. 


157* 


/ 

Sine. 

D.  1". 

Cosine. 

— j 

D.  1\ 

| Tang. 

D.  1\ 

Cotang. 

/ 

0 

9.573575 

5.22 

5.20 

5.20 

5.20 

5.20 

5.18 

5.18 

5.18 

5.17 

5.17 

5.17 

9.967166 

.85 

.85 

.85 

.87 

.85 

.85 

.85 

.87 

.85 

.87 

.85 

9.606410 

6.05 

6.07 

6.05 

6.05 

6.03 

6.05 

6.03 

6.03 

6.03 

6.03 

6.02 

10.393590 

60 

1 

.573888 

.967115 

.606773 

.393227 

59 

2 

.574200 

.967064 

.607137 

.392863 

58 

3 

.574512 

. .967013 

.607500 

.392500 

57  ' 

4 

.574824 

.966961 

.607863 

.392137 

56 

5 

.575136 

.966910 

.608225 

.391775 

55 

6 

.575447 

.966859 

.608588 

.391412 

54 

7 

.575758 

.966808 

.608950 

.391050 

53 

8 

.576069 

.966756 

.609312 

.390688 

52 

9 

.576379 

.966705 

. 609674 

.390326 

51 

10 

.576689 

.966653 

.610036 

.389964 

50 

11 

9.576999 

5.17 

5.15 

5.15 

5.15 

5.15 

5.13 

5.15 

5.13 

5.12 

5.13 

9.966602 

.87 

.85 

.87 

9.610397 

6.03 

6.02 

6.00 

6.02 

6.00 

6.00 

6.00 

6.(0 

6.00 

5.98 

10.389603 

49 

12 

.577309 

.966550 

.610759 

.389241 

48 

13 

.577618 

.966499 

.611120 

.388880 

47 

14 

.577927 

.966447 

.611480 

.388520 

46 

15 

.578236 

.966395 

.85 

.87 

.87 

.87 

.87 

.85 

.611841 

.388159 

45 

16 

.578545 

.966344 

.612201 

.387799 

44 

17 

.578853 

.966292 

.612561 

.387439 

43 

18 

.579162 

.966240 

.612921 

.387079 

42 

19 

.579470 

.966188 

.613281 

.386719 

41 

20 

.579777 

.966136 

.613641 

.386359 

40 

21 

9.580085 

5.12 

5.12 

5.10 

5.12 

5.10 

5.10 

5.08 

5.10 

5.08 

5.08 

9 . 966085 

.87 

.87 

.87 

.88 

.87 

.87 

.87 

.87 

.88 

.87 

9.614000 

5.98 

5.98 

5.98 

5.97 

5.97 

5.9/ 

5.97 

5.97 

5.95 

5.97 

10.386000 

39 

22 

.580392 

.966033 

.614359 

.385641 

38 

23 

.580699 

.965981 

.614718 

.385282 

37 

24 

.581005 

.965929 

.615077 

.384923 

36 

25 

.581312 

.965876 

.615435 

.384565 

35 

26 

.581618 

.965824 

.615793 

.384207 

34 

27 

.581924 

.965772 

.616151 

.383849 

33 

28 

.582229 

.965720 

.616509 

.383491 

32 

29 

.582535 

.965668 

.616867 

.383133 

31 

30 

.582840 

.965615 

.617224 

.382776 

SO 

31 

9.583145 

5.07 

5.08 
5.07 
5.05 
5.07 
5.05 
5.07- 
5.03 
5.05 
5.03 

9.965563 

.87 

.88 

.87 

.88 

.87 

.88 

eft 

9.617582 

5.95 

5.93 

5.95 

5.93 

5.93 

5.93 

5.93 

5.93 

5.92 

5.92 

10.382418 

29 

32 

.583449 

.965511 

.617939 

.382061 

28 

33 

.583754 

.965458 

.618295 

.381705 

27 

34 

.584058 

.965406 

.918652 

.381348 

26 

35 

.584361 

.965353 

.619008 

.380992 

25 

36 

.584665 

.965301 

.619364 

.380636 

24 

37 

.584968 

.965248 

.619720 

.380280 

23 

38 

.585272. 

.965195 

.OO 

.87 

OO 

.620076 

.379924 

22 

39 

.585574 

.965143 

.620432 

.379568 

21 

40 

.585877 

.965090 

.OO 

.88 

.620787 

.379213 

20 

41 

9.586179 

5.05 

5.02 

5.03 

5.02 

5.03 
5.02 
5.00 
5.02 
5.00 
5.00 

9.965037 

qq 

9.621142 

5.92 

5.92 

5.92 

5.90 

5.90 

5.90 

5.90 

5.88 

5.90 

5.88 

10.378858 

19 

42 

.586482 

.964984 

.OO 

.621497 

.378503 

18 

43 

.586783 

.964931 

.OO 

.87 

qq 

.621852 

.378148 

17 

44 

.587085 

.964879 

.622207 

.377793 

16 

45 

.587386 

.964826 

.OO 

QQ 

.622561 

.377439 

15 

46 

.587688 

.964773 

.OO 

QQ 

'.622915 

.377085 

14 

47 

.587989 

.964720 

,oo 

.90 

ce 

.623269 

.376731' 

13 

48 

.588289 

.964666 

.623623 

.376377 

12 

49 

.588590 

.964613 

.OO 

ftft 

.623976 

.376024 

11 

50 

.588890 

.964560 

.OO 

.88 

.624330 

.375670 

10 

51 

9.589190 

4.98 

5.00 

4.98 

4.98 

4.98 

4.97 

4.97 

4.97 

4.97 

9.964507 

ftft 

9.624683 

5.88 
5.87 
■ 5.88 
5.87 
5.87 
5.87 
5.87 
5.87 
5.85 

10.375317 

9 

52 

.589489 

.964454 

.OO 

.90 

QQ 

.625036 

.374964 

8 

53 

.589789 

.964400 

.625388 

.374612 

7 

54 

.590088 

.964347 

.OO 

QQ 

.625741 

.374259 

6 

55 

.590387 

.964294 

.OO 

.90 

QQ 

.626093 

.373907 

5 

56 

.590686 

.964240 

.626445 

.373555 

4 

57 

.590984 

.964187 

.OO 

.90 

ce 

.626797 

.373203 

3 

58 

.591282 

.964133 

.627149 

.372851 

2 

59 

.591580 

.964080 

.OO 

.90 

.627501 

.372499 

1 

60 

9.591878 

9.964026 

9.627852 

10.372148 

0 

/ 

Cosine. 

D.  1". 

| Sine. 

D.  1\  1 

Cotang. 

d.  r. 

Tang. 

' 

220 


J12‘ 


67* 


COSINES,  TANGENTS,  AND  COTANGENTS. 


156< 


$3“ 


J 

Sine. 

D.  1*. 

Cosine. 

d.  r. 

i 

j Tang. 

d.  r. 

Cotang. 

/ 

0 

9.591878 

4.97 

4.95 

4.95 

4.95  | 

4.93 

4.93 

4.93 

4.93 

4.93 

4.92 

4.92 

9.964026 

.90 

9.627852 

5.85 

5.85 

5.85 

5.83 

5.85 

5.83 

5.83 

5.83 

5.82 

5.83 
5.82 

10.372148 

60 

1 

.592176 

.963972 

.628203 

.371797 

59 

2 

.592473 

.963919 

.90 

.90 

.90 

ftft 

; .628554 

.371446 

58 

3 

.592770 

.963865 

.628905 

.371095 

57 

4 

.593067 

.963811 

.629255 

.370745 

56 

5 

.593363 

.963757 

.629606 

.370394 

55 

6 

.593659 

.963704 

.90 

.90 

.90 

.90 

.90 

I .629956 

.370044 

54 

7 

. 593955 

.963650 

| .630306 

.369694 

53 

8 

.594251 

.963596 

. 630656 

.369344 

52 

9 

.594547 

.963542 

1 .631005 

.368995 

51 

10 

.594842 

.963488 

.631355 

.368645 

50 

11 

9.595137 

4.92 

4.92 

4.90 

4.90 

4.90 

4.90 

4.88 

4.90 

4.88 

4.87 

9.963434 

.92 

.90 

.90 

.90 

.90 

.92 

.90 

.92 

.90 

.92 

9.631704 

5.82 

5.82 

5.80 

5.82 

5.80 

5.80 

5.80 

K 7ft 

10.368296 

49 

12 

.595432 

.963379 

.632053 

.367947 

48 

13 

.595727 

.963325 

1 .632402 

.367598 

47 

14 

.596021 

.963271 

( .632750 

.367250 

46 

15 

-.596315 

.963217 

.633099 

.366901 

45 

16 

.596609 

.963163 

.633447 

.366553 

44 

17 

.596903 

.963108 

.633795 

.366205 

43 

18 

.597196 

.963054 

.634143 

.365857 

42 

19 

.597490 

.962999 

.634490 

o.  <o 
5.80 
5.78 

.365510 

41 

20 

.597783 

.962945 

.634838 

.365162 

40 

21 

9.598075 

4.88 

4.87 

4.87 

4.87 

4.87 

4.85 

4.85 

4.85 

4.85 

4.83 

9.962890 

.90 

.92 

.90 

.92 

.92 

.92 

.90 

.92 

.92 

.92 

9.635185 

5.78 

£ 7ft 

10.364815 

39 

22 

.598368 

.962836 

.635532 

.364468 

38 

23 

.598860 

.962781 

.635879 

o . to 

K 7ft 

.364121 

37 

24 

.598952 

.962727 

.636226 

0.  <o 
K 77 

.363774 

36 

25 

.599244 

.962672 

.636572 

O.  ( l 
K 7ft 

.363428 

35 

26 

.599536 

.962617 

.636919 

O.  <o 
K 77 

.363081 

34 

27 

.599827 

.962562 

.637265 

5.77  . 
5.75 

K 77 

.362735 

33 

28 

.600118 

.962508 

.637611 

.362389 

32 

29 

.600409 

.962453 

I .637956 

.362044 

31 

30 

.600700 

.962398 

| .638302 

5.75 

.361698 

30 

31 

9.600990 

4.83 

4.83 

4.83 

4.83 

4.82 

4.82 

4.82 

4.80 

4.82 

4.80 

9.962343 

.92 

.92 

.92 

.92 

.93 

.92 

.92 

.92 

.93 

.92 

9.638647 

K 7K 

10.361353 

29 

32 

.601280 

.962283 

.638992 

D.  ( O 
K 7K 

.361008 

28 

33 

.601570 

.962233 

.639337 

D.  ( D 
K 7K 

.360663 

27 

34 

.601860 

.962178 

.639682 

0.(0 
K 7K 

.360318 

26 

35 

.602150 

.962123 

.640027 

O . 40 

5.73 

. 359973 

25 

36 

.602439 

.962067 

! .640371 

.359629 

24 

37 

.602728 

.962012 

! .640716 

O.  (O 

5.73 

£ 7Q 

.359284 

23 

38 

.603017 

.961957 

.641060 

.358940 

22 

39 

.603305 

.961902 

.641404 

O . i O 

5.72 

5.73 

.358596 

21 

40 

.603594 

.961846 

.641747 

.358253 

20 

41 

9.603882 

4.80 

4.78 

4.80 

A 7ft 

9.961791 

.93 

.92 

.93 

.92 

.93 

.92 

.93 

.93 

.93 

.92 

9.642091 

5.72 

K 70 

10.357909 

19 

42 

.604170 • 

.961735 

.642434 

.357566 

18 

43 

.604457 

.961680 

.642777 

5.72 

5.72 

K 70 

.357223 

17 

44 

.604745 

.961624 

.643120 

.356880 

16 

45 

.605032 

A.  (O 

4r78 

4.78 

A 77 

.961569 

.643463 

.356537 

15 

46 

• .605319 

.961513 

.643806 

O.  (6 

5.70 
k 7 n 

.356194 

14 

47 

.605606 

.961458 

.644148 

.355852 

13 

48 

.605892 

c±.  i i 

4.78 

4.77 

4.77 

.981402 

.644490 

O.  (U 

5.70 

5.70 

5.70 

.355510 

12 

49 

.606179 

.961346 

.644832 

.355168 

11 

50 

.606465 

.961290 

.645174 

.354826 

10 

51 

9.606751 

A 7K 

9.961235 

.93 

.93 

.93  ! 

.93 

.93 

.93 

.93 

.95 

9.645516 

5.68 

5.70 

5.68 

5.68 

5.68 

5.67 

5.68 
5.67 
5.67 

10.354484 

9 

52 

.607036 

A 77 

.961179 

' . 645857 

.354143 

8 

53 

.607322 

i i 

a 7*=; 

.961123 

.646199 

.353801 

7 

54 

.607607 

A 7K 

.961067 

.646540 

.353460 

6 

55 

.607892 

4.  (O 
A 7K 

.961011 

.646881 

.353119 

5 

56 

.608177 

4.73 

4.73- 

4.73 

4.73 

.960955 

.647222 

.352778 

4 

57 

.608461 

.960899 

.647562 

.352438 

3 

58 

.608745 

.960843 

.647903 

.352097 

2 

59 

.609029 

.960786 

.648243 

.351757 

1 

60 

9.609313 

9.960730 

.93 

9.648583 

10.351417 

0 

' 1 Cosine. 

D.  1". 

Sine. 

i D.  1".  j 

| Cotang. 

D.  1". 

| Tang. 

/ 

113* 


221 


66‘ 


24°  TABLE  XII.  LOGARITHMIC  SIXES,  I550 


1 

' 1 Sine. 

! 

D.  1". 

Cosine. 

D.  1". 

: Tang. 

D.  V.  j 

1 

Cotang. 

1 ' 

0 

9.609313 

4.73 
4.72  I 

9.960730 

.93 

.93 

.95 

.93 

.95 

.93 

.95 

.93 

.95 

.95 

.93 

9.648583 

5.67 

5.67 

5.65 

5.67 

5.65 

5.65 

5.65 

5.63 

5.65 

5.63 

5.63 

10.351417 

60 

1 

.609597 

.960674 

1 .648923 

.351077  | 

! 59 

2 

.609880 

.960618 

.649263 

.350737 

i 58 

3 

.610164 

4.72 

4.70 

4.72 

.960561 

.649602 

. 350398 

! 57 

4 

.610417 

.960505 

.649942 

.350058 

i 56 

5 

.610729 

.960448 

.650281 

.349719 

55 

6 

.611012 

.960392 

.650620 

.349380 

j 54 

7 

.611294 

4.70 

.960335 

.650959 

.349041 

53 

8 

.611576 

.960279 

.651297 

.348703 

52 

9 

.611858 

4.70 

4.68 

.960222 

.651636 

.348364 

51 

13 

.612140 

.960165 

.651974 

.348026 

50 

11 

9.612421 

4.68 

4.68 

4.68 

4.68 

4.67 

4.67 

4.67 

4.67 

4.65 

4.65 

9.960109 

.95 

.95 

.95 

.93 

.95 

.95 

.95 

.95 

.97 

.95 

9.652312 

5.63 

5.63 

5.63 

5.62 

5.62 

5.62 

5.62 

5.62 

5.62 

5.60 

10.347688 

49 

12 

.612702 

.960052 

. 652650 

.347350 

48 

13 

.612983 

. 959995 

.652988 

.347012 

47 

14 

.613264 

.959938 

.653326 

.346674 

46 

15 

.613545 

.959882 

.653663 

.346337 

45 

16 

.613825 

.959825 

.654000 

.346000 

44 

17 

.614105 

.959768 

.654337 

.345663 

43 

18 

.614385 

.959711 

.654674 

.345326 

42 

19 

.614665 

.959654 

.655011 

.344989 

41 

20 

.614944 

.959596 

.655348 

.344652 

40 

21 

9.615223 

4.65 

4.65 

4.65 

4.63 

4.63 

4.63 

4.63 

4.63 

4.62 

4.62 

9.959539 

.95 

.95 

.95 

,97 

.95 

.97 

.95 

.97 

.95 

.97 

9.655684 

5.60 

5.60 

5.60 

5.60 

5.60 

5.58 

5.58 

5.58 

5.58 

5.58 

10.344316 

39 

22 

.615502 

.959482 

.656020 

.343980 

38 

23 

.615781 

.959425 

.656356 

.343644 

37 

24 

.616060 

.959368 

.656692 

.343308 

36 

25 

.616338 

.959310 

.657028 

.342972 

35 

26 

.616616 

.959253 

.657364 

.342636 

34 

27 

.616894 

.959195 

.657699 

.342301 

33 

28 

.617172 

.959138 

.658034 

.341966 

32 

29 

.617450 

.959080 

.658369 

.341631 

31 

30 

.617727 

.959023 

.658704 

.341296 

30 

31 

9.618004 

4.62 

4.62 

4.60 

4.60 

4.60 

4.60 

4.60 

4.58 

4.58 

4.58 

9.958965 

.95 

.97 

.97 

.97 

.95 

.97 

.97 

.97 

.97 

.97 

9.659039 

5.57 

5.58 
5.57 
5.57 
5.57 
5.55 
5.57 
5.55 
5.55 
5.55 

10.340961 

29 

32 

.618281 

.958908 

.659373 

.340627 

28 

33 

.618558 

.958850 

.659708 

.340292 

27 

34 

.618834 

.958792 

.660042 

.339958 

26 

35 

.619110 

.958734 

.660376 

.339624 

25 

36 

.619386 

.958677 

.660710 

.339290 

24 

37 

.619662 

.958619 

.661043 

.338957 

23 

38 

.619938 

.958561 

.661377 

.338623 

22 

33 

.620213 

.958503 

.661710 

.338290 

21 

40' 

.620488 

.958445 

.662043 

.337957 

20 

41 

9.620763 

4.58 

4.58 

4.57 

4.57 

4.57 

4.57 

4.55 

4.57 

4.55 

4.55 

9.958387 

.97 

.97 

.97 

.98 

.97 

.97 

.98 

.97 

.97 

.98 

9.662376 

5.55 

5.55 

5.55 

5.53 

5.53 

5.53 

5.53 

5.53 

5.52 

5.53 

10.337624 

19 

42 

.621038 

.958329 

.662709 

.337291 

18 

43 

.621313 

.958271 

.663042 

.336958 

17 

44 

.621587 

.958213 

. 603375 

.336625 

16 

45 

.621861 

.958154 

.663707 

.336293 

15 

46 

.622135 

.958096 

.664039 

.335961 

14 

47 

.622409 

.958038 

.664371 

.335629 

13 

48 

.622682 

.957979 

.664703 

.335297 

12 

49 

.622956 

.957921 

.665035 

.334965 

11 

50 

.623229 

. 957863 

.665366 

.334634 

10 

51 

9.623502 

4.53 

4.55 

4.53 

4.53 

4.53 

4.53 

4.52 

4.52 

4.52 

9.957804 

.97 

Oft 

9.665698 

5.52 

5.52 

5.52 

5.50 

5.52 

5.50 

5.52 

5.50 

5.50 

10.334302 

9 

52 

.623774 

.957746 

.666029 

.333971 

8 

53 

.624047 

.957687 

.VO 

Oft 

.666360 

.333640 

7 

54 

.624319 

.957628 

.'Jo 

.97 

.98 

98 

.666691 

.333309 

6 

55 

.624591 

.957570 

.667(21 

.332979 

5 

56 

. 624863 

.957711 

.667352  1 

.332648 

4 

57 

.625135 

.957452 

'.98 

.97 

.98 

.667682 

.332318 

3 

58 

.625406 

.957393 

.668013 

.331987 

2 

59 

60 

.625677 

9.625948 

.957335 

9.957276 

.668343 

9.668673 

.331657 

10.331327 

1 

0 

/ 

Cosine. 

I d.  r. 

Sine. 

Vr.1 

Cotang. 

D.  1".  I 

Tang. 

M 

114‘ 


222 


05* 


25‘ 


COSINES,  TANGENTS,  AND  COTANGENTS, 


154* 


/ 

Sine. 

D.  r. 

Cosine. 

D.  1". 

Tang. 

D.  1". 

Cotang. 

/ 

0 

9.625948 

4.52 

4.52 

4.50 

4.50 

4.50 

4.50 

4.50 

4.48 

4.48 

4.48 

4.48 

4.48 

4.47 

4.47 

4.47 

4.47 

4.45 

4.47 

4.45 

4.45 

4.45 

4.43 

4.43 

4.45 

4.43 

4.42 

4.43 
4.42 
4.42 
4.42 
4.42 

4.42 
4.40 
4.40 
4.40 
4.40 
4.40 
4.38 
4 38 
4.38 
4.38 

4.37 

4.38 
4.37 
4.37 
4.37 
4.35 
4.37 
4.35 
4.35 
4.35 

4.35 

4.33 

4.33 

4.33 

4.33 

4.33 

4.33 

4.32 

4.32 

9 . 957276 

.98 

.98 

.98 

9.668673 

5.48 

5.50 

5.48 

5.50 

5.48 

5.48 

5.47 

5.48 
5.48 
5.47 
5.47 

5.47 

5.47 

5.45 

5.47 

5.45 

5.47 

5.45 

5.45 

5.43 

5.45 

5.43 

5.45 

5.43 

5.43 

5.42 

5.43 
5.43 
5.42 
5.42 
5.42 

5.42 

5.42 

5.40 

5.42 

5.40 

5.40 

5.40 

5.40 

5.40 

5.38 

5.40 

5.38 

5.38 

5.38 

5.38 

5.37 

5.38 
5.37 
5.37 
5.37 

5.37 

5.37 

5.35 

5.37 

5.35 

5.35 

5.35 

5.35 

5.35 

10.331327 

60 

1 

.626219 

.957217 

.669002  1 

.330998 

59 

2 

.626490 

.957158 

.669332  1 

.330668 

58 

3 

.626760 

.957099 

.669661 

.330339 

57 

4 

.627030 

.957040 

.669991 

.330009 

56 

5 * 

.627300 

.956981 

1.00 

.98 

.670320 

.329680 

55 

6 

.627570 

.956921 

.670649 

.329351 

54 

7 

.627840 

.956862 

.670977 

.329023 

53 

8 

.628109 

.956803 

.671306 

.328694 

52 

9 

.628378 

.956744 

1.00 

.98 

.98 

1.00 

.671635 

.328365 

51 

10 

11 

.628647 

9.628916 

.956684 

9.956625 

.671963 

9.672291 

.328037 

10.327709 

10 

49 

12 

.629185 

. 956566 

.672619 

.327381 

48 

13 

.629453 

.956506 

.672947 

.327053 

47 

14 

.629721 

.956447 

1.00 

1.00 

.673274 

.326726 

46 

15 

.629989 

.956387 

.673602 

.326398 

45 

16 

.630257 

. 956327 

.673929 

.326071 

44 

17 

.630524 

.956268 

1.00 

1.00 

.98 

1.00 

1.00 

1.00 

1.00 

1.00 

1.00 

1.00 

1.00 

.98 

1.00 

1.00 

1.00 

1.02 

1.00 

1.02 

1.00 

1.02 

1.00 

1.02 

1.02 

1.00 

1.02 

1.02 

1.02 

1.02 

1.02 

1.02 

1.02 

1.02 

1.02 

1.02 

1.02 

1.03 

1.02 

.674257 

.325743 

43 

18 

.630792 

.956208 

.674584 

.325416 

42 

19 

.631059 

.956148 

.674911 

.325089 

41 

20 

21 

.631326 

9.631593 

.956089 

9.956029 

.675237 

9.675564 

.324763 

10.324436 

40 

39 

22 

.631859 

.955969 

.675890 

.324110 

38 

23 

.632125 

.955909 

.676217 

.323783 

37' 

24 

.632392 

.955849 

.676543 

.323457 

36 

25 

.632658 

. 955789 

.676869 

.323131 

35 

• 26 

.632923 

.955729 

.677194 

.322806 

34 

27 

.633189 

.955669 

.677520 

.322480 

33 

28 

.633454 

.955609 

.677846 

.322154 

32 

29 

.633719 

.955548 

.678171 

.321829 

31 

30 

31 

.633984 

9.634249 

.955488 

9.955428 

.678496 

9.678821 

.321504 

10.321179 

30 

29 

32 

.634514 

.955368 

.679146 

.320854 

28 

33 

.634778 

.955307 

.679471 

.320529 

27 

34 

.635042 

.955247 

.679795 

.680120 

.320205 

26  ' 

35 

.635306 

.955186 

.319880 

25 

36 

.635570 

.955126 

.680444 

.319556 

24 

37 

. 635834 

. 955065 

.680768 

.319232 

23 

38 

.636097 

.955005 

.681092 

.318908 

22 

39 

.636360 

.954944 

.681416 

.318584 

21 

40 

41 

.636623 

9.636886 

.954883 

9.954823 

.681740 

9.682063 

.318260 

10.317937 

20 

19 

42 

.637148 

.954762 

.682387 

.317613 

18 

43 

.637411 

.954701 

.682710 

.317290 

17 

44 

.637673 

.954640 

.683033 

.316967 

16 

45 

. 637935 

.954579 

.683356 

.316644 

15 

46 

.638197 

.954518 

.683679 

.316321 

14 

47 

.638458 

.954457 

.684001 

.315999 

13 

48 

.638720 

.954396 

.684324 

.315676 

12 

49 

.638981 

.954335 

.684646 

.315354 

11 

50 

51 

.639242 

9.639503 

.954274 

9.954213 

. .684968 
9.685290 

.315032 

10.314710 

10 

9 

52. 

.639764 

.954152 

.685612 

.314388 

8 

53 

.640024 

.954090 

.685934 

.314066 

V 

54 

.640284 

.954029 

.686255 

.313745 

6 

55 

.640544 

.953968 

1 .02 
1.03 
1.02 

.686577 

.313423 

5 

56 

.640804 

.953906 

.686898 

.313102 

4 

57 

.641064 

.953845 

.687219 

.312781 

3 

58 

.641324 

.953783 

1.03 

1.02 

1.03 

.687540 

.312460 

2 

59 

.641583 

.953722 

.687861 

.312139 

1 

60 

9.641842 

9.953660 

9.688182 

10.311818 

0 

' 

I Cosine. 

1 D.  1\  | 

Sine. 

1 D.  1".  1 

Cotang. 

D.  1\ 

1 Tang. 

' 1 

115’  223  64* 


TABLE  XII.  LOGARITHMIC  SIKES. 


153® 


2G° 


' 

Sine. 

D.  1". 

Cosine. 

D.  1\ 

Tang. 

D.  1\ 

Cotang. 

0 

1 

2 

3 

4 

5 
G 

7 

8 
9 

10 

11 

12 

13 

14 

15 

16 
17 
]8 

19 

20 

21 

22 

23 

24 

25 

26 

27 

28 

29 

30 

31 

32 

33 
31 

35 

36 

37 
33 

39 

40 

41 

42 

43 

44 

45 

46 

47 

48 

49 

50 

51 

52 

53 

54 

55 

56 

57 

58 

59 

60 

9.641842 

.642101 

.642360 

.642618 

.642877 

.643135 

.643393 

.643650 

.643908 

.644165 

.644423 

9.644680 

.644936 

.645193 

.645450 

.645706 

.645962 

.646218 

.646474 

.646729 

.646984 

9.647240 
.647494 
.647749 
.648004 
.648258 
.648512 
. 648766 
.649020 
.649274 
.649527 

9.649781 

.650034 

.650287 

.650539 

.650792 

.651044 

.651297 

.651549 

.651800 

.652052 

9.652304 
.652555 
.652806 
.653057 
653308 
.653558 
.653808 
.654059 
.654309 
. 654558 

9.654808 
.655058 
.655307 
.655556 
.655805 
. 656054 
656302 
. 656551 
.656799 

9.657047 

4.32 

4.32 

4.30 

4.32 

4.30 

4.30 

4.28 

4.30 

4.28 

4.30 

4.28 

4.27 

4.28 
4.28 
4.27 
4.27 
4.27 
4.27 
4.25 
4.25 
4.27 

4.23 

4.25 

4.25 

4.23 

4.23 

4.23 

4.23 

4.23 

4.22 

4.23 

4.22 

4.22 

4.20 

4.22 

4.20 

4.22 

4.20: 

4.18 

4.20 

4.20 

4.18 

4.18 

4.18 

4.18 

4.17 

4.17 

4.18 
4.17 
4.15 
4.17 

4.17 

4.15 

4.15 

4.15 

4.15 

4.13 

4.15 

4.13 

4.13 

9.953660 

.953599 

.953537 

.953475 

.953413 

.953352 

.953290 

.953228 

.953166 

.953104 

.953042 

9.952980 

.952918 

.952855 

.952793 

.952731 

.952669 

.952606 

.952544 

.952481 

.952419 

9.952356 

.952294 

.952231 

.952168 

.952106 

.952043 

.951980 

.951917 

.951854 

.951791 

9.951728 

.951665 

.951602 

.951539 

.951476 

.951412 

'.951349 

.951286 

.951222 

.951159 

9.951096 

.951032 

.950968 

.950905 

'.950841 

.950778 

.950714 

.950650 

.950586 

.950522 

9.950458 
.950394 
.950330 
.950266 
.950202 
.950138 
.950074 
.950010 
. 949945 

9.949881 

1.02 

1.03 

1.03 

1.03 

1.02 

1.03 

1.03 

1.03 

1.03 

1.03 

1.03 

1.03 

1.05 

1.03 

1.03 

1.03 

1.05 

1.03 

1.05 

1.03 

1.05 

1.03 
1.05 
1.05 
1.03 
1.05 
1.05 
1.05 
1.05 
1.05 
1.05 
1.05 
1 05 
1.05 
1.05 
1.07 
1.05 
1 05 
1 07 
1 05 
1.05 
1.07 
1.07 
1.05 
1.07 
1 05 
1.07 
1.07 
1.07 
1.07 
1.07 

1.07 

1.07 

1.07 

1.07 

1.07 

1.07 

1.07 

1.08 
1.07 

9.688182 

.688502 

.688823 

.689143 

.689463 

.689783 

.690103 

.690423 

.690742 

.691062 

.691381 

9.691700 

.692019 

.692338 

.692656 

.692975 

.693293 

.693612 

.693930 

.694248 

.694566 

9.694883 

.695201 

.695518 

.695836 

.696153 

.696470 

.696787 

.697103 

.697420 

.697736 

9.698053 

.698369 

.698685 

.699001 

.699316 

.699632 

.699947 

.700263 

.700578 

.700893 

9.701208 
.701523 
.701837 
.702152 
.702466 
.702781 
.703095 
.703409 
. 703722 
.704036 

9.704350 

.704663 

.704976 

.705290 

.705603 

.705916 

.706228 

.706541 

.706854 

9.707166 

5.33 

5.32 

5.33 
5.33 
5.33 
5.33 
5.33 

5.32 

5.33 
5.32 
5.32 

5.32 

5.32 

5.30 

5.32 

5.30 

5.32 

5.30 

5.30 

5.30 

5.28 

5.30 

5.28 

5.30 

5.28 

5.28 

5.28 

5.27 

5.28 

5.27 

5.28 

5.27 

5.27 

5.27 

5.25 

5.27 

5.25 

5.27 

5.25 

5.25 

5.25 

5.25 

5.23 

5.25 

5.23 

5.25 

5.23 

5.23 

5.22 

5.23 
5.23 

5.22 

5.22 

5.23 
5.22 
5.22 
5.20 
5.22 
5.22 
5.20 

10.311818 

.311498 

.311177 

.310857 

.310537 

.310217 

.309897 

.309577 

.309258 

.308938 

.308619 

10.308300 

.307981 

.307662 

.307344 

.307025 

.306707 

.306388 

.306070 

.305752 

.305434 

10.305117 

.304799 

.304482 

.304164 

.303847 

.303530 

.303213 

.302897 

.302580 

.302264 

10.301947 

.301631 

.301315 

.300999 

.300684 

.300368 

.300053 

.299737 

.299422 

.299107 

10.298792 

.298477 

.298163 

.297848 

.297534 

.297219 

.296905 

.296591 

.296278 

.295964 

10.295650 

.295337 

.295024 

.294710 

.294397 

.294084 

.293772 

.293459 

.293146 

10.292834 

\ 

60 

59 

58 

57 

56 

55 

54 

53 

52 

51 

50 

49 

48 

47 

46 

45 

44 

43 

42 

41 

40 

39 

38 

37 

36 

35 

34 

33 

32 

31 

30 

29 

28 

27 

26 

25 

24 

23 

22 

21 

20 

19 

18 

17 

16 

15 

14 

13 

12 

11 

10 

9 

8 

7 

6 

5 

4 

3 

2 

1 

0 

/ t 

Cosine. 

D.  r. 

i Sine.  1 

L.  1". 

1 Cotang. 

i).  r. 

Tang. 

f 

116‘ 


224 


63‘ 


27° 


COSINKS,  TANGENTS,  AND  COTANGENTS. 


152° 


/ 

Sine. 

D.  r. 

Cosine. 

d.  r. 

0 

1 

2 

3 

4 

5 

6 

7 

8 
9 

10 

11 

12 

13 

14 

15 

16 

17 

18 

19 

20 

21 

22 

23 

24 

25 

26 

27 

28 

29 

30 

31 

32 

33 

34 

35 

36 

37 

38 

39 

40 

41 

42 

43 

44 

45 

46 

47 

48 

49 

50 

51 

52 

53 

54 

55 

56 

57 

58 

59 

60 

9.657047 
. 657295 
.657542 
.657790 
.658037 
.658284 
.658531 
.658778 
.659025 
.659271 
659517 

9.659763 

.660009 

.660255 

.660501 

.666746 

.660991 

.661236 

.661481 

.661726 

.661970 

9.662214 

.662459 

.662703 

.662946 

.663190 

.663433 

.663677 

.663920 

.664163 

.664406 

9.664648 

.664891 

.665133 

.665375 

.665617 

.665859 

.666100 

.666342 

.666583 

.666824 

9.667065 

.667305 

.667546 

.667786 

.668027 

.668267 

.668506 

.668746 

.668986 

.669225 

9.669464 

.669703 

.669942 

.670181 

.670419 

.670658 

.670896 

.671134 

.671372 

9.671609 

4.13 

4.12 

4.13 
4.12 
4.12 
4.12 
4.12 
4.12 
4.10 
4.10 
4.10 

4.10 

4.10 

4.10 

4.08 

4.08 

4.08 

4.08 

4.08 

4.07 

4.07 

4.08 
4.07 
4.05 
4.07 
4.05 
4.07 
4.05 
4.05 
4.05 
4.03 

4.05 

4.03 

4.03 

4.03 

4.03 

4.02 

4.03 
4.02 
4.02 
4.02 

4.00 

4.02 

4.00 

4.02 

4.00 

3.98 

4.00 

4.00 

3.98 

3.98 

3.98 

3.98 

3.98 

3.97 

3.98 
3.97 
3.97 
3.97 
3.95 

9.949881 

.949816 

.949752 

.949688 

.949623 

.949558 

.949494 

.949429 

.949364 

.949300 

.949235 

9.949170 

.949105 

.949040 

.948975 

.948910 

.948845 

.948780 

.948715 

.948650 

.948584 

9.948519 

.948454 

.948388 

.948323 

.948257 

.948192 

.948126 

.948060 

.947995 

.947929 

9.947863 

.947797 

.947731 

.947665 

.947600 

.947533 

.947467 

.947401 

.947335 

.947269 

9.947203 

.947136 

.947070 

.947004 

.946937 

.946871 

.946804 

.946738 

.946671 

.946604 

9.946538 

.946471 

.946404 

.946337 

.946270 

.946203 

.946136 

.946069 

.946002 

9.945935 

1.08 

1.07 

1.07 

1.08 
1.08 

1.07 

1.08 
1.08 

1.07 

1.08 
1.08 

1.08 
1.08 
1.08 
1.08 
1.08 
1.08 
1.08 
• 1.08 
1.10 
1.08 

1.08 

1.10 

1.08 

1.10 

1.08 

1.10 

1.10 

1.08 

1.10 

1.10 

1.10 

uo 

1.10 

1.08 

1.12 

1.10 

1.10 

1.10 

1.10 

1.10 

1.12 

1.10 

1.10 

1.12 

1.10 

1.12 

1.10 

1.12 

1.12 

1.10 

1.12 

1.12 

1.12 

1.12 

1.12 

1.12 

1.12 

1.12 

1.12 

/ 

Cosine. 

D.  1". 

Sine. 

1 D.  1°. 

Tang. 

D.  r. 

Cotang. 

I ' 

9.707166 

5.20 

5.20 

5.20 

5.20 

5.20 

5.18 

5.20 

5.18 

5.18 

5.18 

5.18 

10.292834 

60 

.707478 

.292522 

59 

.707790 

.292210 

58 

.708102 

.291898 

57 

.708414 

.291586 

56 

.708726 

.291274 

55 

.709037 

.290963 

54 

.709349 

.290651 

53 

.709660 

.290340 

52 

.709971 

.290029 

51 

.710282 

.289718 

50 

9.710593 

5.18 

5.18 

5.17 

5.18 
5.17 
5.17 
5.17 
5.17 
5.17 
5.17 

10.289407 

49 

.710904 

.289096 

48 

.711215 

.288785 

47 

.711525 

.288475 

46 

.711836 

.288164 

45 

.712146 

.287854 

44 

.712456 

.287544 

43 

.712766 

.287234 

42 

.713076 

.286924 

41 

.713386 

.286614 

40 

9.713696 

5.15 

5.15 

5.17 

5.15 

5.15 

5.15 

5.15 

5.13 

5.15 

5.13 

10.286304 

39 

.714005 

.285995 

38 

.714314 

.285686 

37 

.714624 

.285376 

36 

.714933 

.285067 

35 

.715242 

.284758 

34 

.715551 

.284449 

33 

.715860 

.284140 

32 

.716168 

.283832 

31 

.716477 

.283523 

30 

9.716785 

•5.13 

5.13 

5.13 

5.13* 

5.13 

5.13 

5.12 

5.13 
5.12 
5.12 

5.12 

5.12 

5.12 

5.10 

5.12 

5.10 

10.283215 

29 

.717093 

.282907 

28 

.717401 

.282599 

27 

.717709 

.282291 

26 

.718017 

.281983 

25 

.718325 

.281675 

24 

.718633 

.281367 

23 

.718940 

.281060 

22 

.719248 

.280752 

21 

.719555 

9.719862 

.280445 

10.280138 

20 

19 

.720169 

.279831 

18 

.720476 

.279524 

17 

.720783 

.279217 

16 

.721089 

.278911 

15 

.721396 

.278604 

14 

.721702 

.278298 

13 

.722009 

5.12 

5.10 

5.10 

5.10 

5.08 

5.10 

5.10 

5.08 

5.08 

5.10 

5.08 

5.08 

5.07 

.277991 

12 

.722315 

.277685 

11 

.722621 

9.722927 

.277379 

10.277073 

10 

9 

.723232 

.276768 

8 

.723538 

.276462 

7 

.723844 

.276156 

6 

.724149 

.275851 

5 

.724454 

.275546 

4 

.724760 

.275240 

3 

. 725065 

.274935 

2 

.725370 

.274630 

1 

9.725674 

10.274326 

0 

Cotang. 

d.  r. 

Tang. 

/ 

62° 


117 


225 


28' 


TABLE  XII.  LOGARITHMIC/  SINES 


151' 


/ 

0 

1 

2 

3 

4 

5 

6 

7 

8 
9 

10 

11 

12 

13 

14 

15 

16 

17 

18 

19 

20 

21 

22 

23 

24 

25 

26 

27 

28 

29 

30 

31 

32 

33 

34 

35 

36 

37 

38 

39 

40 

41 

42 

43 

44 

45 

46 

47 

48 

49 

50 

51 

52 

53 

54 

55 

56 

57 

58 

59 

60 

Sine. 

D.  1\ 

Cosine. 

d.  r. 

Tang. 

D.  1". 

Cotang. 

/ 

9.671609 

.671847 

.672084 

.672321 

.672558 

.672795 

.673032 

.673268 

.673505 

.673741 

.673977 

9.674213 

.674448 

.674684 

.674919 

.675155 

.675390 

.675624 

.675859 

.676094 

.676328 

9.676562 

.676796 

.677030 

.677264 

.677498 

.677731 

.677964 

.678197 

.678430 

.678663 

9.678895 

.679128 

.679360 

.679592 

.679824 

.680056 

.680288 

.680519 

.680750 

.680982 

9.681213 

.681443 

.681674 

.681905 

.682135 

.682365 

.682595 

.682825 

.683055 

.683284 

9.683514 

.683743 

.683972 

.684201 

.684430 

.684658 

.684887 

.685115 

.685343 

9.685571 

3.97 

3.95 

3.95 

3.95 

3.95 

3.95 

3.93 

3.95 

3.93 

3.93 

3.93 

3.92 

3.93 

3.92 

3.93 
3.92 
3.90 
3.92 
3.92 
3.90 
3.90 

3.90 

3.90 

3.90 

3.90 

3.88 

3.88 

3.88 

3.88 

3.88 

3.87 

3.88 
3.87 
3.87 
3.87 
3.87 
3.87 
3.85 
3.85 
3.87 
3.85 

3.83 

3.85 

3.85 

3.83 

3.83 

3.83 

3.83 

3.83 

3.82 

3.83 

3.82 

3.82 

3.82 

3.82 

3.80 

3.82 

3.80 

3.80 

3.80 

9.945935 

.945868 

.945800 

.945733 

.945666 

.945598 

.945531 

.945464 

.945396 

.945328 

.945261 

9.945193 

.945125 

.945058 

.944990 

.944922 

.944854 

.944786 

.944718 

.944650 

.944582 

9.944514 

.944446 

.944377 

.944309 

.944241 

.944172 

.944104 

.944036 

.943967 

.943899 

9.943830 

.943761 

.943693 

.943624 

.943555 

.943486 

.943417 

.943348 

.943279 

.943210 

9.943141 

.943072 

.943003 

.942934 

.942864 

.942795 

.942726 

.942656 

.942587 

.942517 

9.942448 

.942378 

.942308 

.942239 

.942169 

.942099 

.942029 

.941959 

.941889 

9.941819 

1.12 

1.13 

1.12 

1.12 

1.13 

1.12 

1.12 

1.13 

1.13 

1.12 

1.13 

1.13 

1.12 

1.13 

1.13 

1.13 

1.13 

1.13 

1.13 

1.13 

1.13 

1.13 

1.15 

1.13 

1.13 

1.15 

1.13 

1.13 

1.15 

1.13 

1.15 

1.15 

1.13 

1.15 

1.15 

1.15 

1.15 

1.15 

1.15 

1.15 

1.15 

1.15 

.1.15 

1.15 

1.17 

1.15 

1.15 

1.17 

1.15 

1.17 

1.15 

1.17 

1.17 

1.15 

1.17 

1.17 

1.17 

1.17 

1.17 

1.17 

9.725674 

.725979 

.726284 

.726588 

.726892 

.727197 

.727501 

.727805 

.728109 

.728412 

.728716 

9.729020 

.729323 

.729626 

! 730233 
.730535 
.730838 
.731141 
.731444 
.731746 

9.732048 
.732351 
.732653 
.732955 
.733257 
. 733558 
.733860 
.734162 
.734463 
.734764 

9.735066 
.735367 
.735668 
.735969 
.736269 
.736570 
. 736870 
.737171 
.737471 
.737771 

9.738071 

.738371 

.738671 

.738971 

.739271 

.739570 

.739870 

.740169 

.740468 

.740767 

9.741066 

.741365 

.741664 

.741962 

.742261 

.742559 

.742858 

.743156 

.743454 

9.743752 

5.08 

5.08 

5.07 

5.07 

5.05 

5.07 

5.07 

5.07 

5.05 

5.07 

5.07 

5.05 

5.05 

5.05 

5.07 

5.03 

5.05 

5.05 

5.05 

5.03 

5.03 

5.05 

5.03 

5.03 

5.03 

5.02 

5.03 
5.03 
5.02 

5.02 

5.03 

5.02 

5.02 

5.02 

5.00 

5.02 

5.00 

5.02 

5.00 

5.00 

5.00 

5.00 

5.00 

5.00 

5.00 

4.98 

5.00 

4.98 

4.98 

4.98 

4.98 

4.98 

4.98 

4.97 

4.98 

4.97 

4.98 
4.97 
4.97 
4.97 

10.274326 

.274021 

.273716 

.273412 

.273108 

.272803 

.272499 

.272195 

.271891 

.271588 

.271284 

K).  270980 
.270677 
.270374 
.270071 
.269767 
.269465 
.269162 
.268859 
.268556 
.268254 

10. 267952 
.267649 
.267347 
.267045 
.266743 
.266442 
.266140 
.265838 
.265537 
.265236 

10.264934 

.264633 

.264332 

.264031 

.263731 

.263430 

.263130 

.262829 

.262529 

.262229 

10.261929 

.261629 

.261329 

.261029 

.260729 

.260430 

.260130 

.259831 

.259532 

.259233 

10.258934 

.258635 

.258336 

.258038 

.257739 

.257441 

.257142 

.256844 

.256546 

10.256248 

60 

59 

58 

57 

56 

55 

54 

53 

52 

51 

50 

49 

48 

47 

46 

45 

44 

43 

42 

41 

40 

39 

38 

37 

36 

35 

34 

33 

32 

31 

30 

29 

28 

27 

26 

25 

24 

23 

22 

21 

20 

19 

18 

17 

16 

15 

14 

13 

12 

11 

10 

9 

8 

7 

6 

5 

4 

3 

2 

1 

0 

/ 

Cosine. 

D.  1". 

Sine. 

I).  1".  ! 

Cotang. 

D.  1". 

Tang.  | ' 

118* 


226 


6T 


COSINES,  TANGENTS,  AND  COTANGENTS. 


150‘ 


29° 


' 

Sine. 

D.  1".  ! 

Cosine. 

D.  1". 

Tang. 

D.  1\ 

Cotang. 

/ 

0 

1 

2 

3 

4 

5 

6 

7 

8 
9 

10 

11 

12 

13 

14 

15 

16 

17 

18 

19 

20 

21 

22 

23* 

24 

25 

26 

27 

28 

29 

30 

31 

32 

33 

34 

35 

36 

37 

38 

39 

40 

41 

42 

43 

44 

45 

46 

47 

48 

49 

50 

51 

52 

53 

54 

55 

56 

57 

58 

59 

60 

9.685571 
.685799 
.686027 
.686254 
.686482 
. 686709 
.686936 
.687163 
.687389 
.687616 
.687843 

9. 688069 
.688295 
.688521 
.688747 
.688972 
.689198 
.689423 
.689648 
.689873 
.690098 

9.690323 

.690548 

.690772 

.690996 

.691220 

.691444 

.691668 

.691892 

.692115 

.692339 

9.692562 

.692785 

.693008 

.693231 

.693453 

.693676 

.693898 

.694120 

.694342 

.694564 

9.694786 

.695007 

.695229 

.695450 

.695671 

.695892 

.696113 

.696334 

.696554 

.696775 

9.696995 

.697215 

.697435 

.697654 

.697874 

.698094 

.698313 

.698532 

.698751 

9.698970 

3.80 

3.80 

3.78 

3.80 

3.78  I 

3.78  1 

3.78 

3.77  1 

3.78 
3.78  ; 
3.77 

3.77 

3.77 

3.77 

3.75 

3.77 

3.75 

3.75 

3.75 

3.75 

3.75 

3.75 

3.73 

3.73 

3.73 

3.73 

3.73 

3.73 

3.72 

3.73 
3.72 

3.72 

3.72 

3.72 

3.70 

3.72 

3.70 

3.70 

3.70 

3.70 

3.70 

3.68 

3.70 

3.68 

'3.68 

3.68 

3.68 

3.68 

3.67 

3.68 
3.67 

3.67 

3.67 

3.65 

3.67 

3.67 

3.65 

3.65 

3.65 

3.65 

9.941819 

.941749 

.941679 

.941609 

.941539 

.941469 

.941398 

.941328 

.941258 

.941187 

.941117 

9.941046 

.940975 

.940905 

.940834 

.940763 

.940693 

.940622 

.940551 

.940480 

.940409 

9.940338 

.940267 

.940196 

.940125 

.940054 

.939982 

.939911 

.939840 

.939768 

.939697 

9.939625 

.939554 

.939482 

.939410 

.939339 

.939267 

.939195 

.939123 

.939052 

.938980 

9.938908 

.938836 

.938763 

.938691 

.938619 

.938547 

.938475 

.938402 

.938330 

.938258 

9.938185 

.938113 

.938040 

.937967 

.937895 

.937822 

.937749 

.937676 

.937604 

9.937531 

1.17 

1.17 

1.17 

1.17 

1.17 

1.18 
1.17 

1.17 

1.18 

1.17 

1.18 

1.18 

1.17 

1.18 
1.18 

1.17 

1.18 
1.18 
1.18 
1.18 
1.18 

1.18 

1.18 

1.18 

1.18 

1.20 

1.18 

1.18 

1.20 

1.18 

1.20 

1.18 
1.20 
1 20 
1.18 
1.20 
1.20 
1.20 
1.18 
1.20 
1.20 
1.20 
1.22 
1.20  | 
1.20  j 
1.20 
1.20 
1.22 
1.20 
1.20 
1.22 

1.20 

1.22 

1.22 

1.20 

1.22 

1.22 

1.22 

1.20 

1.22 

9.743752 

.744050 

.744348 

.744645 

.744943 

.745240 

.745538 

.745835 

.746132 

.746429 

.746726 

9.747023 

.747319 

.747616 

.747913 

.748209 

.748505 

.748801 

.749097 

.749393 

.749689 

9.749985 
j .750281 
.750576 
.750872 
.751167 
.751462 
.751757 
.752052 
.752347 
.752642 

9.752937 
.753231 
.753526 
.753820 
.754115 
.754409 
.754703 
.754997 
.755291 
. 755585 

9.755878 
.756172 
.756465 
.756759 
.757052 
.757345 
.757638 
.757931 
.758224 
‘ .758517 

9.758810 

.759102 

.759395 

.759687 

.759979 

.760272 

.760564 

.760856 

.761148 

9.761439 

4.97 

4.97 

4.95 

4.97 

4.95 

4.97 

4.95 

4.95 

4.95 

4.95 

4.95 

4.93 

4.95 

4.95 

4.93 

4.93 

4.93 

4.93 

4.93 

4.93 

4.93 

4.93 

4.92 

4.93 
4.92 
4.92 
4.92 
4.92 
4.92 
4.92 
4.92 

4.90 
4 92 
4.90 
4.92 
4.90 
4.90 
4.90 
4.90 
4.90 
4.88 

4.90 

4.88 

4.90 

4.88 

4.88 

4.88 

4.88 

4.88 

4.88 

4.88 

4.87 

4.88 
4.87 

4.87 

4.88 
4.87 
4.87 
4.87 
4.85 

10.256248 

.255950 

.255652 

.255355 

.255057 

.254760 

.254462 

.254165 

.253868 

.253571 

.253274 

10.252977 

.252681 

.252384 

.252087 

.251791 

.251495 

.251199 

.250903 

.250607 

.250311 

10.250015 

.249719 

.249424 

.249128 

.248833 

.248538 

.248243 

.247948 

.247653 

.247358 

10.247063 

.246769 

.246474 

.246180 

.245885 

.245591 

.245297 

.245003 

.244709 

.244415 

10.244122 

.243828 

.243535 

.243241 

.242948 

.242655 

.242362 

.242069 

.241776 

.241483 

10.241190 

.240898 

.240605 

.240313 

.240021 

.239728 

.239436 

.239144 

.238852 

10.238561 

60 

59 

58 

57 

56 

55 

54 

53 

52 

51 

50 

49 

48 

47 

46 

45 

44 

43 

42 

41 

40 

39 

38 

37 

36 

35 

34 

33 

32 

31 

30 

29 

28 

27 

26 

25 

24 

23 

21 

21 

20 

19 

18 

17 

16 

15 

14 

13 

12 

11 

10 

9 

8 

7 

6 

5 

4 

3 

2 

1 

0 

! ' 

! Cosine. 

D.  1". 

Sine. 

D.  1".  i 

! Cotang. 

D.  1". 

1 Tang. 

' 

1199  60® 

227 


30° 


TABLE  XII.  LOGARITHMIC  SIXES. 


149< 


/ 

Sine. 

D.  1". 

Cosine. 

D.  1*. 

Tang. 

D.  1". 

Co  tang. 

/ 

0 

9.698970 

3.65 

3.63 

3.65 

3.63 

3.63 

3.63 

3.63 

3.63 

3.62 

3.63 
3.62 

9.937531 

1.22 

1.22 

1.22 

1.23 

1.22 

1.22 

1.22 

1.22 

1.23 

1.22 

1.23 

9.761439 

4.87 

4.87 

4.85 

4.87 

4.85 

4.85 

4.85 

4.85 

4.85 

4.85 

4.85 

10.238561 

60 

1 

.699189 

.937458 

.761731 

.238269 

59 

2 

.699407 

.937385 

.762023 

.237977 

58 

3 

.699626 

.937312 

.762314 

.237686 

57 

4 

.699844 

.937238 

.762606 

.237394 

56 

5 

.700062 

.937165 

.762897 

.237103 

55 

C 

. 700280 

.937092 

.763188 

.236812 

54 

7 

.700498 

.937019 

.763479 

.236521 

53 

8 

.700716 

.936946 

.763770 

.236230 

52 

9 

.700933 

.936872 

.764061 

.235939 

51 

10 

.701151 

.936799 

.764352 

.235648 

50 

11 

9.701368 

3.62 

3.62 

3.62 

3.62 

3.60 

3.62 

3.60 

3.60 

3.60 

3.60 

9.936725 

1.22 

1.23 

1.22 

1.23 

1.23 

1.22 

1.23 

1.23 

1.23 

1.23 

9.764643 

4.83 
4.85 
4.83 
4.85 
4.83 
4 83 
4.83 
4.83 
4.83 
4.83 

10.235357 

49 

12 

. 7^1585 

.936652 

.764933 

.235067 

48 

13 

.701802 

.936578 

.765224 

.234776 

47 

14 

.702019 

.936505 

.765514 

.234486 

46 

15 

.702236 

.936431 

.765805 

.234195 

45 

16 

.702452 

.936357 

.766095 

.233905 

44 

17 

.702669 

.936284 

.766385 

.233615 

43 

18 

.702885 

.936210 

.766675 

.233325 

42 

19 

.703101 

.936136 

.766965 

.233035 

41 

20 

.703317 

.936062 

.767255 

.232745 

40 

21 

9.703533 

3.60 

3.58 

3.58 

3.60 

3.58 

3.58 

3.58 

3.57 

3.58 
3.57 

9.935988 

1.23 

1.23 

1.23 

1.23 

1.23 

1.25 

1.23 

1.23 

1.25 

1.23 

9.767545 

4.82 

4.83 
4.83 
4.82 
4.82 

4.82 

4.83 
4.82 
4.80 
4.82 

10.232455 

39 

22 

.703749 

.935914 

.767834 

.232166 

38  - 

23 

.703964 

.935840 

.768124 

.231876 

37 

24 

.704179 

.935766 

.768414 

.231586 

36 

25 

.704395 

.935692 

.768703 

.231297 

35 

26 

.701610 

.935618 

.768992 

.231008 

34 

27 

.704825 

.935543 

.769281 

.230719 

33 

28 

.705040 

.935469 

.769571 

.230429 

32 

29 

.705254 

.935395 

.769860 

.230140 

31 

30 

.705469 

.935320 

.770148 

.229852 

30 

31 

9.705683 

3.58 

3.57 

3.57 

3.55 

3.57 

3.57 

3.55 

Q.KK 

9.935246 

1.25 
1.23 
1.25 
1.23 
1.25 
1.25 
1.25 
1.23 
i ok 

9,770437 

4.82 

4.82 

4.80 

4.82  ' 

4.80 

4.80 

4.82 

4.80 

4.80 

4.80 

10.229563 

29 

32 

.705898 

.935171 

.770726 

.229274 

28 

33 

.706112 

.935097 

.771015 

.228985 

27 

34 

.706326 

.935022 

.771303 

.228697 

26 

35 

.706539 

.934948 

.771592 

.228408 

25 

36 

.706753 

.934873 

.771880 

.228120 

24 

37 

.706967 

.934798 

.772168 

.227832 

23 

38 

.707180 

.934723 

.772457 

.227543 

22 

39 

.707393 

O t)J 

3.55 

3.55 

.934649 

.772745 

.227255 

21 

40 

.707606 

.934574 

J . /wt) 

1.25 

.773033 

.226967 

20 

41 

9.707819 

3.55 

3.55 

3.55 

3.53 

3.53 

3.53 

3.53 

3.53 

3.53 

3.52 

9.934499 

1.25 

1.25 

1.25 

1.25 

1.27 

1.25 

1.25 

1.25 

1.27 

1.25 

9.773321 

4.78 

4.80 

4.80 

4.78 

4.80 

4.78 

4.78 

4.80 

4.78 

4.78 

10.226679 

19 

42 

.708032 

.934424 

.773608 

.226392 

18 

43 

.708245 

.934349 

.773896 

.226104 

17 

44 

.708458 

.934274 

.774184 

.225816 

16 

45  j 

.708670 

.934199 

.774471 

.225529 

15 

46  ! 

47  i 

48 

.708882 

.709094 

.709306 

.934123 

.934048 

.933973 

.774759 

.775046 

.775333 

.225241 

.224954 

.224667 

14 

13 

12 

49  | 

.709518 

.933898 

.775621 

.224379 

11 

50 

.709730 

.933822 

.775908 

.224092 

10 

51 

9.709941 

3.53 

3.52 

3.52 

3.52 

3.52 

3.52 

3.52 

3.50 

3.50 

9.933747 

1.27 

1.25 

1.27 

1.25 

1.27 

1.27 

1.27 

1.27 

1.25 

9.776195 

4.78 

4.77 

4.78 
4 78 

10.223805 

9 

52 

.710153 

.933671 

.776482 

.223518 

8 

53 

.710364 

.933596 

. 7767'68 

.223232 

7 

54 

.710575 

.933520 

.777055 

.222945 

6 

55 

.710786 

933445 

.777342 

4 77 

.222658 

5 

56 

.710997 

.933369 

.777628 

4 78 

.222372 

4 

57 

.711208 

.933293 

.777915 

4 77 

.222085 

3 

58 

.711419 

.933217 

.778201 

4'.78 

4.77 

.221799 

2 

59 

.711629 

.933141 

.778488 

.221512 

1 

60 

9.711839 

9.933066 

9.778774 

10.221226 

0 

/ 

1 Cosine. 

D.  1\ 

Sine. 

I).  1\ 

Cotang. 

D.  1". 

Tang.  1 

/ 

120‘ 


228 


31 


COSINES,  TANGENTS,  AND  COTANGENTS. 


148‘ 


t 

Sine. 

d.  r. 

0 

9.711839 

3.52 

3.50 

3.48 

3.50 

3.50  | 

3.48 

3.50 

3.48 

3.48 

3.48 

3.48 

1 

.712050 

2 

.712260 

3 

4 

5 

.712469 

.712679 

.712889 

6 

.713098 

7 

.713308 

8 

9 

.713517 

.713726 

10 

.713935 

11 

9.714144 

3.47 

3.48 

3.47 

3.48 
3.47 
3.47 
3.47 
3.45 
3.47 
3.45 

12 

.714352 

13 

.714561 

14 

.714769 

15 

.714978 

16 

.715186 

17 

.715394 

18 

.715602 

19 

.715809 

20 

.716017 

21 

9.716224 

3.47 

3.45 

3.45 

3.45 

3.43 

3.45 

3.45 

3.43 

3.43 

3.43 

22 

.716432 

23 

.716639 

24 

.716846 

25 

.717053 

26 

.717259 

27 

.717466 

28 

.717673 

29 

.717879 

30 

.718085 

31 

9.718291 

3.43 

3.43 

3.43 

3.42 

3.43 
3.42 
3.42 
3.42 
3.42 
3.42 

32 

.718497 

33 

.718703 

34 

.718909 

35 

.719114 

36 

.719320 

37 

.719525 

38 

.719730 

39 

.719935 

40 

.720140 

41 

9.720345 

3.40 
3.42 
3.40 
' 3.40 
3.40 
3.40 
3.40 
3.40 
3.38 
3.40 

42 

.720549 

43 

.720754 

44 

.720958 

45 

.721162 

46 

.721366 

47 

.721570 

48 

49 

.721774 

.721978 

50 

.722181 

51 

9.722385 

3.38 

3.38 

3.38 

3.38 

3.38 

3.38 

3.37 

3.37 

3.38 

52 

.722588 

53 

54 

.722791 

.722994 

55 

.723197 

56 

.723400 

57 

.723603 

58 

.723805 

59 

.724007 

60 

9.724210 

/ 

Cosine. 

D.  1". 

Cosine. 


D.  1". 


9.938066 

.932990 

.982914 

.932838 

.932762 

.932085 

.932609 

.932533 

.932457 

.932380 

.932304 


1.27 

1.27 

1.27 

1.27 

1.28 
1.27 
1.27 

1.27 

1.28 
1.27 
1.27 


9.932228 

.932151 

.932075 

.931998 

.931921 

.931845 

.931768 

.931691 

.931614 

.931537 

9.931460 

.931383 

.931306 

.931229 

.931152 

.931075 

.930998 

.930921 

.930843 

.930766 

9.930688 

.930611 

.930533 

.930456 

.930378 

.930300 

.930223 

.930145 

.930067 

.929989 

9.929911 

.929833 

.929755 

.929677 

.929599 

.929521 

.929442 

.929364 

.929286 

.929207 

9.929129 

.929050 

.928972 

.928893 

.928815 

.928736 

.928657 

.928578 

.928499 

9.928420 


1.28 

1.27 

1.28 
1.28 

1.27 

1.28 
1.28 
1.28 
1.28 
1.28 

1.28 

1.28 

1.28 

1.28 

1.28 

1.28 

1.28 

1.30 

1.28 

1.30 

1.28 

1.30 

1.28 

1.30 

1.30 

1.28 

1.30 

1.30 

1.30 

1.30 

1.30 
1 30 
1.30 
1 30 
1 30 
1 32 
1.30 
1 30 
1.32 
1.30 
1 32 
1.30 
1.32 
1.30 
1.32 
1.32 
1.32 
1.32 
1.32 


Sine,  j D.  1". 


Tang. 

D.  1". 

Cotang. 

/ 

9.778774 

A W 

10.221226 

60 

.779060 

A W 

.220940 

59 

.779346 

e±.  i i 

A W 

.220054 

58 

.779632 

e±.  ( i 

A 77 

.220368 

57 

.779918 

e±.  t ( 

A >7K 

.220082 

56 

.780203 

A . <0 
A 77 

.219797 

55 

.780489 

4.  I ( 
A 77 

.219511 

54 

.780775 

.219225 

53 

.781060 

4 .75 

.218940 

52 

.781346 

4.77 

.218654 

51 

.781631 

4.  40 

4.75 

.218369 

50 

9.781916 

A WK 

10.218084 

49 

.782201 

<± . 4 0 
A >7K 

.217799 

48 

.782486 

A VK 

.217514 

47 

.782771 

4 . <0 
A VK 

.217229 

46 

.788056 

4.  id 
A VK 

.216944 

45 

.783341 

4.  <0 
A 

.216659 

44 

.783626 

4.  id 
A 

- .216874 

43 

.783910 

4.  iO 
A r/K 

.216090 

42 

.784195 

4.  i D 
A 

.215805 

41 

.784479 

4.40 

4.75 

.215521 

40 

9.784764 

A *7Q 

10.215236 

39 

.785048 

4.4  0 
A 

.214952 

38 

.785332 

4.  40 
A >7Q 

.214668 

37 

.785616 

4.  (O 

.214384 

36 

.785900 

4.73 

A CQ 

.214100 

35 

.786184 

4.  (O 

A C'Q 

.213816 

34 

.786468 

4.  40 
A 7Q 

.213532 

33 

.786752 

4.40 

A 

.213248 

32 

.787036 

4.  /O 
A VO 

.212964 

31 

.787319 

4.4/^ 

4.73 

.212681 

30 

9.787603 

A 70 

10.212397 

29 

. 787'886 

4.  (A 
A 7Q 

.212114 

28 

.788170 

4.  40 
A 70 

.211830 

27 

.788453 

4. 

A 70 

.211547 

26 

.788736  • 

4.  4/0 
A 70 

.211264 

25 

.789019 

4.  14 
A 70 

.210981 

24 

.789302 

4.  4/0 
A 70 

.210698 

23 

.789585 

4.  4/0 
A 70 

.210415 

22 

.789868 

4.4/0 
A 70 

.210132 

21 

.790151 

4.4  0 

4.72 

.209849 

20 

9.790434 

A 741 

10.209566 

19 

.790716 

4.  4 U 
A 70 

.209284 

1 18 

.790999 

4.40 
A 741 

.209001 

j 17 

.791281 

4.  4U 
A 70 

.208719 

1 16 

.791563 

4.  4 U 
A 70 

.208437 

15 

.791846 

4.  4/W 
A 70 

.208154 

1 i 

.792128 

4.  4 U 
A 70 

.207872 

13 

.792410 

4.  4 U 
A 70 

.207590 

12 

.792692 

4.  4U 
A 70 

.207308 

11 

.792974 

4 . 4 U 

4.70 

.207026 

10 

9.793256 

A 70 

10.206744 

9 

.793538 

4.  4U 
A Aft 

.206462 

8 

.793819 

4.  Oo 
A 70 

.206181 

7 

.794101 

4.  4U 
A 70 

.205899 

6 

.794383 

4.  4 U 
A Aft 

.205617 

5 

.794664 

4.00 
A 70 

.2U5336 

4 

.794916 

4.  4U 
A Aft 

.205054 

3 

.795227 

4.00 
A Aft 

.204773 

2 

.795508 

4.00 
A Aft 

.204492 

1 

9. 79578 J 

4 . Oo 

10.204211 

0 

Cotang. 

D.  1". 

Tang. 

/ 

121 


229 


58‘ 


32‘ 


TABLE  X1L  LOGARITHMIC  SINES. 


147< 


' 

Sine. 

D.  r. 

Cosine. 

D.  1'. 

Tang. 

D.  1\ 

Cotang. 

' 

0 

1 

2 

3 

4 

5 

6 

7 

8 
9 

10 

11 

12 

13 

14 

15 

16 

17 

18 

19 

20 

21 

22 

23 

24 

25 

26 

27 

28 

29 

30 

31 

32 

33 

34 

35 

36 

37 

38 

39 

40 

41 

42 

43 

44 

45 

46 

47 

48 

49 

50 

51 

52 

53 

54 

55 

56 

57 

58 

59 

60 

9.724210 

.724412 

.724614 

.724816 

.725017 

.725219 

.725420 

.725622 

.725823 

.726024 

.726225 

9.726426 

.726626 

.726827 

.72/027 

.727228 

.727428 

.727628 

.727828 

.728027 

.728227 

9.728427 

.728626 

.728825 

.729024 

.729223 

.729422 

.729621 

.729820 

.730018 

.730217 

9 730415 
.730613 
.730811 
.731009 
.731206 
.731404 
.731602 
.731799 
. 731996 
.732193 

9.732390 

.732587 

.732784 

.732980 

.733177 

.733373 

.733569 

.733765 

.733961 

.734157 

9.734353 
.734549 
.734744 
.734939 
.735135 
. 735330 
.735525 
.735719 
.735914 
9.736109 

3.37 

3.37 

3.37 

3.35 

3.37 

3.35 

3.37 

3.35 

3.35 

3.35 

3.35 

3.33 
3.35 
3.33 
3.35 
3.33 
3 33 
3.33 

3.32 

3.33 
3.33 

3.32 

3.32 

3.32 

3.32 

3.32 

3.32 

3.32 

3.30 

3.32 

3.30 

3.30 

3.30 

3.30 

3.28 

3.30 

3.30 

3.28 

3.28 

3.28 

3.28 

3.28 

3.28 

3.27 

3.28 
3.27 
3.27 
3.27 
3.27 
3.27 
3.27 

3.27 

3.25 

3.25 

3.27 

3.25 

3.25 

3.23 

3.25 

3.25 

9.928420 
.928342 
.928263 
.923183 
.928104 
.928025 
.927946 
.927867 
. 927787 
.927708 
.927629 

9.927549 

.927470 

.927390 

.927310 

.927231 

.927151 

.927071 

.926991 

.926911 

.926831 

9.926751 
.926671 
.'926591 
.926511 
.926431 
.926351 
. 926270 
.926190 
.926110 
.926029 

9.925949 
.925868 
.925788 
.925707 
.925626 
.925545 
.925465 
.925384 
. 925303 
.925222 

9.925141 

.925060 

.924979 

.924897 

.924816 

.924735 

.924654 

.924572 

.924491 

.924409 

9.924328 

.924246 

.924164 

.924083 

.924001 

.923919 

.923837 

.923755 

.923673 

9.923591 

1.30 

1.32 

1.33 
1.32 
1.32 
1.32 

1.32 

1.33 
1.32 

1.32 

1.33 

1.32 

1.33 
1.33 

1.32 

1.33 
1.33 
1.33 
1.33 
1.33 
1.33 

1.33 

1.33 

1.33 

1.33 

1.33 

1.35 

1.33 

1.33 

1.35 

1.33 

1.35 
1.33 
1.35 
1.35 
1.35 
1.33 
1.35 
1.35 
• 1.35 
1.35 

1.35 

1.35 

1-37 

1.35 

1.35 

1.35 

1.37 

1.35 

1.37 

1.35 

1.37 
1.37 
1.35 
1.37 
1.37 
1.37 
1 37 
1.37 

_^lI 

9.795789 

.796070 

.796351 

,796632 

.796913 

.797194 

.797474 

.797755 

.798036 

.798316 

.798596 

9.798877 

.799157 

.799437 

.799717 

.799997 

.800277 

.800557 

.800836 

.801116 

.801396 

9.801675 

.801955 

.802234 

.802513 

.802792 

.803072 

.803351 

.803630 

.803909 

.804187 

9.804466 

.804745 

.805023 

.805302 

.805580 

.805859 

.806137 

.806415 

.806693 

.806971 

9.807249 

.807527 

.807805 

.808083 

.808361 

.808638 

.808916 

.809193 

.809471 

.809748 

9.810025 

.810302 

.810580 

.810857 

.811134 

.811410 

.811687 

.811964 

.812241 

9.812517 

4.68 

4.68 

4.68 

4.68 

4.68 

4.67 

4.68 
4.68 
4.67 

4.67 

4.68 

4.67 

4.67 

4.67 

4.67 

4.67 

4.67 

4.65 

4.67 

4.67 

4.65 

4.67 

4.65 

4.65 

4.65 

4.67 

4.65 

4.65 

4.65 

4.63 

4.65 

4.65 
4.63 
4 65 
4.63 
4.65 
4 63 
4.63 
4.63 
4.63 
4.63 

4.63 

4.63 

4.63 

4.63 

4.62 

4.63 

4.62 

4.63 
4.62 
4.62 

4.62 

4.63 
4.62 
4.62 
4.60 
4.62 
4.62 
4.62 
4.60 

10.204211 

.203930 

.203649 

.203368 

.203087 

.202806 

.202526 

.202245 

.201964 

.201684 

.201404 

10.201123 

.200843 

.200563 

.200283 

.200003 

.199723 

.199443 

.199164 

.198884 

.198604 

10.198325 

.198045 

.197766 

.197487 

.197208 

.196928 

.196649 

.196370 

.196091 

.195813 

10.195534 
. 195255 
.194977 
.194698 
.194420 
. 194141 
.193863 
.193585 
.193307 
.193029 

10.192751 
.192473 
.192195 
.191917 
. 191639 
.191362 
.191084 
.190807 
.190529 
.190252 

10.189975 
.189698 
. 189420 
.189143 
.188866 
.188590 
.188313 
. ISSOOli 
.187759 

10.187483 

60 

59 

58 

57 

56 

55 

54 

53 

52 

51 

50 

49 

48 

47 

46 

45 

44 

43 

42 

41 

40 

39 

38 

37 

36 

35 

34 

33 

32 

31 

30 

29 

28 

27 

26 

25 

24 

23 

22 

21 

20 

19 

18 

17 

16 

15 

14 

13 

12 

11 

10 

9 

8 

6 

5 

4 

3 

2 

1 

0 

' | Cosine. 

D.  1".  1 

Sine. 

d.  r.  |l 

Cotang. 

D.  1". 

Tang. 

9 

122‘ 


230 


57' 


s: 


COSINES,  TANGENTS,  AND  COTANGENTS. 


146< 


' 

Sine. 

D.  1". 

Cosine. 

D.  r. 

Tang. 

D.  1\ 

Cotang. 

i 

0 

1 

2 

3 

4 

5 

6 

7 

8 
9 

10 

11 

12 

13 

14 

15 

16 

17 

18 

19 

20 

21 

22 

23 

24 

25 

26 

27 

28 

29 

30 

31 

32 

33 

34 

35 

36 

37 

38 

39 

40 

41 

42 

43 

44 

45 

46 

47 

48 

49 

50 

51 

52 

53 

54 

55 

56 

57 

58 

59 

60 

9.736109 
. 736303 
.736498 
.736692 
.736886 
.737080 
.737274 
.737467 
.737661 
.737855 
.738048 

9.738241 

.738434 

.738627 

.738820 

.739013 

.739206 

.739398 

.739590 

.739783 

.739975 

9.740167 
.740359 
. 740550 
.740742 
.740934 
.741125 
.741316 
.741508 
.741699 
.741889 

9.742080 

.742271 

.742462 

.742652 

.742842 

.743033 

.743223 

.743413 

.743602 

.743792 

9.743982 

.744171 

.744361 

.744550 

.744739 

.744928 

.745117 

.745306 

.745494 

.745683 

9.745871 

.746060 

.746248 

.746436 

.746624 

.746812 

.746999 

.747187 

.747374 

9.747562 

3.23 

3.25 

3.23 

3.23 

3.23 

3.23 

3.22 

3.23 

3.23 

3.22 

3.22 

3.22 

3.22 

3.22 

3.22 

3.22 

3.20 

3.20 

3.22 

3.20 

3.20 

3.20 

3.18 

3.20 

3.20 

3.18 

3.18 

3.20 

3.18 

3.17 

3.18 

3.18 

3.18 

3.17 

3.17 

3.18 
3.17 
3.17 
3.15 
3.17 
3.17 

3.15 
3.17 
3.15 
3.15 
S.  15 
3.15 
3.15 
3.13 
3.15 
3.13 

3.15 

3.13 

3.13 

3.13 

3.13 

3.12 

3.13 

3.12 

3.13 

9.923591 

.923509 

.923427 

.923345 

.923263 

.923181 

.923098 

.923016 

.922933 

.922851 

.922768 

9.922686 

.922603 

.922520 

.922438 

.922355 

.922272 

.922189 

.922106 

.922023 

.921940 

9.921857 

.921774 

.921691 

.921607 

.921524 

.921441 

.921357 

.921274 

.921190 

.921107 

9.921023 

.920939 

.920856 

.920772 

.920688 

.920604 

.920520 

.920436 

.920352 

.920268 

9.920184 

,920099 

.920015 

.919931 

.919846 

.919762 

.919677 

.919593 

.919508 

.919424 

9.919339 

.919254 

.919169 

.919085 

.919000 

.918915 

.918830 

.9187'45 

.918659 

9.918574 

1.37 

1.37 

1.37 

1.37 

1.37 

1.38 

1.37 

1.38 

1.37 

1.38 

1.37 

1.38 
1.38 

1.37 

1.38 
1.38 
1.38 
1.38 
1.38 
1.38 
1.38 

1.38 

1.38 

1.40 

1.38 

1.38 

1.40 

1.38 

1.40 

1.38 

1.40 

1.40 

1.38 

1.40 

1.40 

1.40 

1.40 

1.40 

1.40 

1.40 

1.40 

1.42 

1.40 

1.40 

1.42 

1.40 

1.42 

1.40 

1.42 

1.40 

1.42 

1.42 

1.42 

1.40 

1.42 

1.42 

1.42 

1.42 

1.43 
1.42 

9.812517 

.812794 

.813070 

.813347 

.813623 

.813899 

.814176 

.814452 

.814728 

.815004 

.815280 

9.815555 

.815831 

.816107 

.816382 

.816658 

.816933 

.817209 

.817484 

.817759 

.818035 

9.818310 

.818585 

.818860 

.819135 

.819410 

.819684 

.819959 

.820234 

.820508 

.820783 

9.821057 

.821332 

.821606 

.821880 

.822154 

.822429 

.822703 

.822977 

.823251 

.823524 

9.823798 

.824072 

.824345 

.824619 

.824893 

.825166 

.825439 

.825713 

.825986 

.826259 

9.826532 

.826805 

.827078 

.827351 

.827624 

.827897 

.828170 

.828442 

.828715 

9.828987 

4.62 

4.60 

4.62 

4.60 

4.60 

4.62 

4.60 

4.60 

4.60 

4.60 

4.58 

4.60 

4.60 

4.58 

4.60 

4.58 

4.60 

4.58 

4.58 

4.60 

4.58 

4.58 

4.58 

4.58 

4.58 

4.57 

4.58 
4.58 

4.57 

4.58 

4.57 

4.58 
4.57 
4.57 

4.57 

4.58 
4.57 
4.57 
4.57 
4.55 
4.57 

4.57 

4.55 

4.57 

4.57 

4.55 

4.55 

4.57 

4.55 

4.55 

4.55 

4.55 

4.55 

4.55 

4.55 

4.55 

4.55 

4.53 

4.55 

4.53 

10.187483 
.187206 
.186930 
. 186653 
.186377 
.186101 
.185824 
.185548 
.185272 
. 184996 
.184720 

10.184445 

.184169 

.183893 

.183618 

.183342 

.183067 

.182791 

.182516 

.182241 

.181965 

10.181690 
.181415 
.181140 
.180865 
. 180590 
.180316 
.180041 
.179766 
.179492 
.179217 

10.178943 
.178668 
.178394 
.178120 
.177846 
.177571 
.177*97 
.177023 
.176749 
. 176476 

10.176202 
.175928 
.175655 
.175381 
. 175107 
.174834 
.174561 
.174287 
.174014 
.173741 

10.173468 
.173195 
.172922 
.172649 
. 172376 
.172103 
.171830 
.171558 
.171285 

10.171013 

60 

59 

58 

57 

56 

55 

54 

53 

52 
51 
50 

49 
48 
47 
46 
• 45 
44 
43 
42 
41 
40 

39 

38 

37. 

36 

35 

34 

53 
32 
31 
30 

29 

28 

27 

26 

25 

24 

23 

22 

21 

20 

19 

18 

17 

16 

15 

14 

13 

12 

11 

10 

9 

8 

7 

6 

5 

4 

3 

2 

1 

0 

' 

Cosine. 

D.  1". 

Sine. 

D.  1". 

Cotang. 

d.  r. 

Tang. 

' 

123° 


231 


56‘ 


34' 


TABLE  XII.  LOGARITHMIC  SIKES, 


145' 


/ 

Sine. 

d.  r. 

0 

1 

2 

3 

9.747562 

.747749 

.747936 

.748123 

3.12 

3.12 

3.12 

3.12 

3.12 

3.10 

3.12 

3.10 

3.12 

3.10 

3.10 

4 

.748310 

5 

6 

.748497 

.748683 

7 

8 

.748870 

.749056 

9 

.749243 

10 

.749429 

11 

9.749615 

3.10 
3.10 
3.08 
3.10 
3.08 
3.10 
3.08 
3.08 
. 3.08 
3.08 

12 

.749801 

13 

.749987 

14 

.750172 

15 

.750358 

16 

.750543 

17 

18 

.750729 

.750914 

19 

.751099 

20 

.751284 

21 

9.751469 

3.08 

3.08 

3.07 

3.08 
3.07 
3.07 
3.07 
3.07 
3.07 
3.07 

22 

.751654 

23 

.751839 

24 

.752023 

25 

.752208 

26 

.752392 

27 

.752576 

28 

.752760 

29 

.752944 

30 

.753128 

31 

9.753312 

3.05 

3.07 

3.07 

3.07 

3.05 

3.05 

3.05 

3.05 

3.03 

3.05 

32 

.753495 

33 

.753679 

34 

.753862 

35 

.754046 

36 

.754229 

37 

.754412 

38 

.754595 

39 

.754778 

40 

.754960 

41 

9.755143 

3. 05 
3.03 
3.03 
3.03 
3.03 
3.03 
3.03 
3.03 
3.03 
3.02 

42 

.755326 

43 

.755508 

44 

.755690 

45 

.755872 

46 

.756054 

47 

48 

.756236 

.756418 

49 

. 756600 

50 

.756782 

51 

9.756963 

3.02 
3 03 
3.02 
3.02 
3.02 
3.02 
3.0.0 
3.02 
3.00 

52 

.757144 

53 

.757326 

54 

.757507 

55 

.757688 

56 

.757869 

57 

.758050 

58 

.758230 

59 

.758411 

60 

9.758591 

/ 

Cosine.  1 

D.  1\ 

Cosine. 


9.918574 
.918489 
.918404 
.918318 
.918233 
.918147 
.918062 
.917976 
. 917891 
.917805 
.917719 

9.917634 
.917548 
.917462 
.917376 
: 917290 
.917204 
.917118 
. 9171)32 
.916946 
.916859 

9.916773 

.916687 

.916600 

.916514 

.916427 

.916341 

.916254 

.916167 

.916081 

.915994 

9.915907 

.915820 

.915733 

.915646 

.915559 

.915472 

.915385 

.915297 

.915210 

.915123 

9.915035 

.914948 

.914860 

.914773 

.914685 

.914598 

.914510 

.914422 

.914334 

.914246 

9.914158 

.914070 

.913982 

.913894 

.913806 

.913718 

.913630 

913541 

.913453 

9.913365 


Sine. 


D.  r. 


1.42 

1.42 

1.43 

1.42 

1.43 

1.42 

1.43 

1.42 

1.43 
1.43 
1:42 


1.43 

1.43 

1.43 

1.43 

1.43 

1.43 

1.43 

1.43 

1.45 

1.43 


1.43 

1.45 

1.43 

1.45 

1.43 

1.45 

1.45 

1.43 

1.45 

1.45 

1.45 

1.45 

1.45 

1.45 

1.45 

1.45 

1.47 

1.45 

1.45 

1.47 

1.45 

1.47 

1.45 

1.47 

1.45 

1.47 

1.47 

1.47 

1.47 

1.47 

1.47 

1.47 

1.47 

1.47 

1.47 

1.47 

1.48 
1.47 
1.47 


Tang. 


D.  1". 


Cotang. 


9.828987 

.829260 

.829532 

.829805 

.830077 

.830349 

.830621 

.830893 

.831165 

.831437 

.831709 


4.55 

4.53 

4.55 


4.53 

4.53 

4.53 

4.53 

4.53 

4.53 

4.53 


10.171013 
. 170740 
.170468 
.170195 


60 

59 

58 

57 


.169923 

.169651 

.169379 

.169107 

.168835 

.168563 

.168291 


56 

55 

54 

53 

52 

51 

50 


9.831981 

.832253 

.832525 

' .832796 
.833068 
.833339 
.833611 
.833882 
.834154 
.834425 

9.834696 
.834967 
.835238 
. 835509 
.835780 
.836051 
.836322 
.836593 
.836864 
.837134 

9.837405 

.837675 

.837946 

.838216 

.838487 

.838757 

.839027 

.839297 

.839568 

.839838 

9.840108 

.840378 

.840648 

.U0917 

.841187 

.841457 

.841727 

.841996 

.812266- 

.842535 

9.842805 

.843074 

.843343 

.843612 

.843882 

.844151 

.844420 

.844689 

.844958 

9.845227 


4.53 

4.53 

4.52 

4.53 

4.52 

4.53 

4.52 

4.53 
4.52 
4.52 

4.52 
4.52 
4.52 
4.52 
4.52 
. 4.52 
4.52 
4.52 
4.50 
4.52 

4.50 

4.52 

4.50 

4.52 

4.50 

4.50 

4.50 

4.52 

4.50 

4.50 

4.50 

4.50 

4.48 

4.50 

4.50 

4.50 

4.48 

4.50 

4.48 

4.50 

4.48 

4.48 

4.48 

4.50 

4.48 

4.48 

4.48 

4.48 

4.48 


10.168019 

49 

.167747 

48 

.167475 

47 

.167204 

46 

.166932 

45 

.166661 

44 

.166389 

43 

.166118 

42 

.165846 

41 

. 165575 

40 

10.165304 

39 

.165033 

38 

.164762 

37 

.164491 

36 

.164220 

35 

.163949 

34 

.163678 

33 

.163407 

32 

.163136 

31 

.162866 

30 

10.162595 

29 

.162325 

28 

.162054 

27 

.161784 

26 

.161513 

25 

.161243 

24 

.160973 

23 

.160703 

22 

.160432 

21 

.160162 

20 

10.159892 

19 

.159622 

18 

.159352 

17 

.159083 

16 

.158813 

15 

.158543 

14 

.158273 

13 

.158004 

12 

. 157734 

11 

.157465 

10 

10.157195 

9 

156926 

8 

.156657 

7 

.156388 

6 

.156118 

5 

.155849 

4 

.155580 

3 

.155311 

2 

.155042 

1 

10.154773  ! 

0 

I D.  1".  1 1 Cotang.  I D.  1".  I Tang. 


124‘ 


232 


55' 


COSINES,  TANGENTS,  AND  COTANGENTS.  144* 


' 

Sine. 

d.  r. 

Cosine. 

| 

D.  1*. 

Tang. 

D.  1". 

Cotang.  | ' 

1 

0 

1 

2 

3 

4 

5 

6 

7 

8 
9 

10 

11 

12 

13 

14 

15 

16 

17 

18 

19 

20 

21 

22 

23 

24 

25 

26 

27 

28 

29 

30 

31 

32 

33 

34 

35 

36 

37 

38 

39 

40 

41 

42 

43 

44 

45 

46 

47 

48 

49 

50 

51 

52 

53 

54 

55 

56 

57 

58 

59 

60 

9.758591 

.758772 

.758952 

.7.59132 

.759312 

.759492 

.759672 

.759853 

.760031 

.760211 

.760390 

9 . 760569 
.760748 
.760927 
.761106 
.761285 
.761464 
.761642 
.761821 
.761999 
.762177 

9.762356 

.762534 

.762712 

.762889 

.763067 

.763245 

.763422 

.763600 

.763777 

.763954 

9.764131 

.764308 

.764485 

.764662 

.764838 

.765015 

.765191 

.765367 

.765544 

.765720 

9.765896 

.766072 

.766247 

.766423 

.766598 

.766774 

.766949 

.767124 

.767300 

.767475 

9.767649 

.767824 

.767999 

.768173 

.768348 

.768522 

.768697 

.768871 

.769045 

9.769219 

3.02 

3.00 

3.00 

3.00 

3.00 

3.00 

3.00 

2.98 

3.00 

2.98 

2.98 

2.98 

2.98 

2.98 

2.98 

2.98 

2.97 

2.98 
2.97 

2.97 

2.98 

2.97 

2.97 

2.95 

2.97 

2.97 

2.95 

2.97 

2.95 

2.95 

2.95 

2.95 

2.95 

2.95 

2.93 

2.95 

2.93 

2.93 

2.95 

2.93 

2.93 

2.93 

2.92 

2.93 
' 2.92 

2.93 

2.92 

2.92 

2.93 
2.92 
2.90 

2.92 

2.92 

2.90 

2.92 

2.90 

2.92 

2.90 

2.90 

2.90 

9.913365 

.913276 

.913187 

.913099 

.913010 

.912922 

.912833 

.912744 

.912655 

.912566 

.912477 

9.912388 

.912299 

.912210 

.912121 

.912031 

.911942 

.911853 

.911763 

.911674 

.911584 

9.911495 

.911405 

.911315 

.911226 

.911136 

.911046 

-.910956 

.910866 

.910776 

.910686 

9.910596 

.910506 

.910415 

.910325 

.910235 

.910144 

.910054 

.909963 

.909873 

.909782 

9.909691 

.909601 

.909510 

.909419 

.909328 

.909237 

.909146 

.909055 

.908964 

.908873 

9.908781 
.908690 
.908599 
.908507 
.908416 
.908324 
.908233 
.908141 
| .908019 

l 9.907958 

1.48 

1.48 

1.47 

1.48 

1.47 

1.48 
1.48 
1.48 
1.48 
1.48 
1.48 

1.48 

1.48 

1.48 

1.50 

1.48 

1.48 

1.50 

1.48 

1.50 

1.48 

1.50 

1.50 

1.48 

1.50 

1.50 

1.50 

1.50 

1.50 

1.50 

1.50 

1.50 

1.52 

1.50 

1.50 

1.52 

1.50 

1.52 

1.50 

1.52 

1.52 

1.50 

1.52 

1.52 

1.52 

1.52 

1.52 

1.52 

1.52 

1.52 

1.53  * 
1.52 

1.52 

1.53 

1.52 

1.53 

1.52 

1.53 
1.53 
1.52 

9.845227 

.845496 

.845764 

.846033 

.846302 

.846570 

.846839 

.847108 

.847376 

.847644. 

.847913 

9.848181 

.848449 

.848717' 

.848986 

.849254 

.849522 

.849790 

.850057 

.850325 

.850593 

9.850861 

.851129 

.851396 

.851664 

.851931 

.852199 

.852466 

.852733 

.853001 

.853268 

9.853535 
.853802 
.854069 
.854336 
.854603 
.854870 
.855137 
.855404 
.855671 
.855938 
9.856204 
.856471 
. 856737 
.857004 
.857270 
. 857537 
.857803 
.858069 
.858336 
.858602 

9.858868 

.859134 

.859400 

.859666 

.859932 

.860198 

.860464 

.860730 

.860995 

9.861261 

4.48 

4.47 

4.48 
4.48 

4.47 

4.48 
4.48 
4.47 

4.47 

4.48 
4.47 

4.47 

4.47 

4.48 
4.47 
4.47 
4.47 
4.45 
4.47 
4.47 
4.47 

'4.47 

4.45 

4.47 

4.45 

4.47 

4.45 

4.45 

4.47 

4.45 

4.45 

4.45 
4.45 
4.45 
4.45 
4.45 
4.45 
4.45 
4. .45 
4.45 
4.43 

4.45 

4.43 

4.45 

4.43 

4.45 

4.43 

4.43 

4.45 

4.43 

4.43 

4.43 

4.43 

4.43 

4.43 

4.43 

4.43 

4.43 

4.42 

4.43 

■ 

10.154773 

.154504 

.154236 

.153967 

.153698 

.153430 

.153161 

.152892 

.152624 

.152356 

.152087 

10.151819 

.151551 

.151283 

.151014 

.150746 

.150478 

.150210 

.149943 

.149675 

.149407 

10.149139 

.148871 

.148604 

.148336 

.148069 

.147801 

.147534 

.147267 

.146999 

.146732 

10.146465 

.146198 

.145931 

.145664 

.145397 

.145130 

.144863 

.144596 

.144329 

.144062 

10.143796 
.143529 
.143263 
.142996 
.142730 
. 142463 
.142197 
.141931 
.141664 
.141398 

10.141132 

.140866 

.140600 

.140334 

.140068 

.139802 

.139586 

.139270 

.139005 

10.138739 

60 

59 

58 

57 

56 

55 

54 

53 

52 

51 

50 

49 

48 

47 
46 
45 
44 

48 
42 
41 
40 

39 

38 

37 

36 

35 

34 

33 

32 

31 

30 

29 

28 

27 

26 

25 

24 

23 

22 

21 

20 

19 

15  . 

17 

16 
15 
14 
13 
12 
11 
10 

9 

8 

6 

5 

4 

3 

1 2 
1 
0 

/ 

Cosine. 

D.  I".' 

[ | Sine. 

D.  1". 

Cotang. 

D.  1\ 

| Tang. 

j ' 

125 


283 


54< 


36 


TABLE  XII.  LOGARITHMIC  SltfES, 


143* 


' 

Sine. 

D.  1". 

Cosine. 

D.  1". 

Tang. 

d.  r. 

Cotang. 

' 

0 

1 

2 

3 

4 

5 
C 

7 

8 
9 

10 

11 

12 

13 

14 

15 

16 

17 

18 

19 

20 

21 

22 

23 

24 

25 

26 

27 

28 

29 

30 

31 

32 

33 

34 

35 

36 

37 

38 

39 

40 

41 

42 

43 

44 

45 

46 

47 

48 

49 

50 

51 

52 

53 

54 

55 

56 

57 

58 

59 

60 

9.769219 

.769393 

.769566 

.769740 

.769913 

.770087 

.770260 

.770433 

.770606 

.770779 

.770952 

9.771125 

.771298 

.771470 

.771643 

.771815 

.771987 

.772159 

.772331 

.772503 

.772675 

9.772847 

.773018 

.773190 

.773361 

.773533 

.773704 

.773875 

.774046 

.774217 

.774388 

9.774558 
.774729 
.774899 
.775070 
.775240 
.775410 
.775580 
. 775750 
.775920 
.776090 

9 . 776259 
.776429 
.776598 
.776768 
.776937 
.777106 
.777275 
.777444 
.777613 
.777781 

9.777950 

.778119 

.778287 

.778455 

.778624 

.778792 

.778960 

.779128 

.779295 

9.779463 

2.90 

2.88 

2.90 

2.88 

2.90 

2.88 

2.88 

2.88 

2.88 

2.88 

2.88 

2.88 

2.87 

2.88 
2.87 
2.87 
2.87 
2.87 
2.87 
2.87 
2.87 
2.85 
2.87 
2.85 
2.87 
2.85 
2.85 
2.85 
2.85 
2.85 
2.83 

2.85 

2.83 

2.85 

2.83 

2.83 

2.83 

2.83 

2.83 

2.83 

2.82 

2.83 

2.82 

2.83 

2.82 

2.82 

2.82 

2.82 

2.82 

2.80 

2.82 

2.82 

2.80 

2.80 

2.82 

2.80 

2.80 

2.80 

2.78 

2.80 

9.907958 

.907866 

.907774 

.907682 

.907590 

.907498 

.907406 

.907314 

.907222 

.907129 

.907037 

9.906945 
.906852 
.906760 
.906667 
. 906575 
.906482 
.906389 
.906296 
.906204 
.906111 

9.906018 

.905925 

.905832 

.905739 

.905645 

.905552 

.905459 

.905366 

.905272 

.905179 

9.905085 

.904992 

.904898 

.904804 

.904711 

.904617 

.904523 

.904429 

.904335 

.904241 

9.904147 

.904053 

.903959 

.903864 

.903770 

.903676 

.903581 

.903487 

.903392 

.903298 

9.903203 

.903108 

.903014 

.902919 

.902824 

.902729 

.902634 

.902539 

.902444 

9.902349 

1.53 

1.53 

1.53 

1.53 

1.53 

1.53 

1.53 

1.53 

1.55 

1.53 

1.53 

1.55 

1.53 

1.55 

1.53 

1.55 

1.55 

1.55 

1.53 

1.55 

1.55 

1.55 

1.55 

1.55 

1.57 

1.55 

1.55 

1.55 

1.57 

1.55 

1.57 

1.55 

1.57 

1.57 

1.55 

1.57 

1.57 

1.57 

1.57 

1.57 

1.57 

1.57 

1.57 

1.58 
1.57 

1.57 

1.58 

1.57 

1.58 

1.57 

1.58 

1.58 

1.57 

1.58 
1.58 
1.58 
1.58  j 
1.58 
1.58 
1.58 

1 

9.861261 
.861527 
.861792 
.862058 
.862323 
.862589 
.862854 
.863119 
.863385 
. 863650 
.863915 

9.864180 
.864445 
.864710 
.864975 
.865240 
.865505 
. 865770 
.866035 
.866300 
.866564 

9.866829 

.867094 

.867’358 

.867623 

.867887 

.868152 

.868416 

.868680 

.868945 

.869209 

9.869473 
.8697  37 
.870001 
.870265 
.870529 
.870793 
.871057 
.871321 
.871585 
.871849 

9.872112 
.872376 
.872640 
.872903 
.873167 
.873430 
.873694 
.873957 
i .874220 
.874484 

9.874747 

.875010 

.875273 

.875537 

.875800 

.876063 

.876326 

.876589 

.876852 

9.877114 

4.43 

4.42 

4.43 

4.42 

4.43 
4.42 

4.42 

4.43 
4.42 
4.42 
4.42 

4.42 
4.42 
4.42 
4.42 
4.42 
4.42 
4.42 
4.42 
4.40 
4.42 
4.42 
4.40 
4:42 
4.40 
4.42 
4.40 
4.40 
4 42 
4.40 
4.40 

4.40 

4.40 

4.40 

4.40 

4.40 

4.40 

4.40 

4.40 

4.40 

4.38 

4.40 

4.40 

4.38 

4.40 

4.38 

4.40 

4.38 

4.38 

4.40 

4.38 

4.38 

4.38 

4.40 

4.38 

4.38 

4.38, 

4.38 

4.38 

4.37 

10.138739 

.138473 

.138208 

.137942 

.137677 

.137411 

.137146 

.136881 

.136615 

.136350 

.136085 

10.135820 
.135555 
.135290 
.135025 
.134760 
. 134495 
.134230 
. 133965 
.133700 
.133436 

10.133171 
.132906 
.132642 
.132377 
.132113 
.131848 
.131584 
. 131320 
. 131055 
.130791 

10.1,30527 

.130263 

.129999 

.129735 

.129471 

.129207 

.128943 

.128679 

.128415 

.128151 

10.127888 

.127624 

.127360 

.127097 

.126833 

.126570 

.126306 

.126043 

.125780 

.125516 

10.125253 
.124990 
.124727 
.124463 
.124200 
.123937 
. 123674 
.123411 
. 123148 

10.122886 

60 

59 

58 

57 

56 

55 

54 

53 

52 

51 

50 

49 

48 

47 

46 

45 

44 

43 

42 

41 

40 

39 

38 

37 

36 

35 

34 

33 

32 

31 

30 

29 

28 

27 

26 

25 

24 

23 

22 

21 

20 

19 

18 

17 

16 

15 

14 

13 

12 

11 

10 

9 

8 

7 

6 

5 

4 

3 

0 

1 
0 

' 

Cosine. 

D.  1\  1 

i Sine. 

D.  1".  i 

Cotang,  i 

D.  1".  1 

Tang. 

/ 

53* 


126c 


234 


S7' 


COSINES,  TANGENTS,  AND  COTANGENTS. 


142° 


Sine. 


9.779463 

.779631 

.779798 


.780133 

.780300 

.780467 

.780634 

.780801 

.780968 

.781134 

9.781301 

.781468 

.781634 

.781800 

.781966 

.782132 

.782298 

.782464 

.782630 

.782796 

9.782961 

.783127 


.783458 

.783623 

.783788 

.783953 

.784118 

.784282 

.784447 

9.784612 

.784776 

.784941 

.785105 


.785433 

.785597 

.785761 

.785925 

.786089 

9.786252 

.786416 

.786579 

.786742 

.786906 

.787069 

.787232 

.787395 

.787557 

.787720 

9 . 787883 
.788045 
.788208 
.788370 
.788532 
.788694 
.788856 
.789018 
.789180 
9.789342 

Cosine. 


D.  1". 


2.80 

2.78 

2.80 

2.78 

2.78 

2.78 

2.78 

2.78 

2.78 

2.77 

2.78 

2.78 

2.77 

2.77 

2.77 

2.77 

2.77 

2.77 

2.77 

2.77 

2.75 

2.77 

2.75 

2.77 

2.75 

2.75 

2.75 

2.75 

2.73 

2.75 

2.75 

2.73 

2.75 

2.73 

2.73 

2.73 

2.73 

2.73 

2.73 

2.73 

2.72 

2.73 
2.72 

2.72 

2.73 
'2.72 
2.72 
2.72 
2.70 
2.72 
2.72 

2.70 

2.72 

2.70 

2.70 

2.70 

2.70 

2.70 

,2.70 

2.70 

D 1". 


Cosine. 

d.  r. 

Tang. 

D.  1". 

Cotang. 

/ 

9.902349 

1.60 
1.58 
1.58 
1.60 
1.58 
1.60 
1.58 
1 .GO 
1.58 
1.60 
1.60 

9.877114 

4.38 

4.38 

4.38 

4.37 

4.38 
4.38 

4.37 

4.38 

4.37 

4.38 
4.37 

10.122886 

60 

. 902253 

.877377 

. 122623 

59 

.902158 

.877640 

.122360 

58 

.902063 

.877903 

.122097 

57 

.901967 

.878165 

.121835 

56 

.901872 

.878428 

.121572 

55 

.901776 

.878691 

.121309 

54 

.901681 

.878953 

.121047 

53 

. 901585 

.879216 

.120784 

52 

.901490 

.879478  - 

. 1205>, 

51 

.901394 

.879741 

.120259 

50 

9.901298 

1.60 
1.60 
1.60 
1.60 
1.60 
1.60 
1.60 
. 1.62 
1.60 
1 60 

9.880003 

4.37 

4.38 
4.37 
4.37 

4.37 

4.38 
4.37 
4.37 
4.37 
4.37 

10.119997 

49 

.901202 

.880265 

.119735 

48 

.901106 

.880528 

.119472 

47 

.901010 

.880790 

.119210 

46 

.900914 

.881052 

.118948 

45 

.900818 

.881314 

.118686 

44 

.900722 

.881577 

.118423 

43 

.900626 

.881839 

.118161 

42 

.900529 

.882101 

. 117899 

41 

.900433 

.882363 

.117637 

40 

9.900337 

1.62 

1.60 

1.62 

1.60 

1.62 

1.62 

1.62 

1.60 

1.62 

1.62 

9.882625 

4.37 

4.35 

4.37 

4.37 

4.37 

4.37 

4.35 

4.37 

4.35 

4.37 

10.117375 

39 

.900240 

.882887 

.117113 

38 

.900144 

.883148 

.116852 

37 

.900047 

.883410 

.116590 

36 

.899951 

.883672 

.116328 

35 

.899854 

.883934 

.116066 

34 

.899757 

.884196 

.115804 

33 

.899660 

.884457 

.115543 

32 

.899564 

.884719 

.115281 

31 

.899467 

.884980 

.115020 

30 

9.899370 

1.62 

1.62 

1.63 

1.62 

1.62 

1.62 

1.63 

1.62 

1.63 

1.62 

9.885242 

10.114758 

29 

.899273 

.885504 

4.0* 

4.35 

4.35 

4.37 

4.35 

4.37 

4.35 

4.35 

4.35 

4.35 

.114496 

28 

.899176 

.885765 

.114235 

27 

.899078 

.886026 

.113974 

26 

.898981 

.886288 

.113712 

25 

.898884 

.886549 

.113451 

24 

.898787 

.886811 

.113189 

23 

.898689 

. 887072 

.112928 

22 

.898592 

.887333 

.112667 

21 

.898494 

.887594 

.112406 

20 

9.898397 

1.63 

1.62 

1.63 

1.63 

1.63 

1.63 

1.63 

1.63 

1.63 

1.63 

9.887855 

4.35 

4.37 

4.35 

4.35 

4.35 

4.33 

4.35 

4.35 

4.35 

4.35 

10.112145 

19 

.898299 

.888116 

.111884 

18 

.898202 

.888378 

.111622 

17 

.898104 

.888639 

.111361 

16 

.898006 

.888900 

.111100 

15 

.897908 

.889161 

.110839 

14 

.897810 

.889421 

.110579 

13 

.897712 

.889682 

.110318 

12 

.897614 

.889943 

.110057 

1 11 

.897516 

.890204 

.109796 

; 10 

9.897418 

1.63 

1.63 

1.65 

1.63 

1.65 

1.63 

1.65 

1.63 

1.65 

9.890465 

4.33 
4.35 
4.35 
4.33 
4.35 
4.33 
4.35 
4. ,33 
4.35 

10.109535 

9 

.897320 

.890725 

. 109275 

8 

.897222 

.890986 

.109014 

7 

.897123 

.897025 

.891247 

.891507 

. 108753 
.108493 

6 

5 

.896926 

.891768 

.108232 

4 

.896828 

.892028 

.107972 

3 

.896729 

.892289 

.107711 

2 

.896631 

892549 

.107451 

1 

9.896532 

9.892810 

10.107190 

* 0 

Sine. 

ID.  1". 

i Cotang. 

1 D.  1". 

Tang.  ' 

127* 


235 


52s> 


38< 


TABLE  XII.  LOGARITHMIC  SIKES, 


141 


/ 

Sine. 

D.  1". 

Cosine. 

D.  1". 

Tang. 

D.  1\ 

Cotang. 

/ 

0. 

9.789342 

2.70 

2.68 

2.70 

2.68 

2.68 

2.68 

2.68 

2.68 

2.68 

2.68 

2.68 

9.896532 

1.65 

1.63 

1.65 

1.65 

1.65 

1.65 

1.65 

1.65 

1.67 

1.65 

1.65 

9.892810 

4.33 

4.35 

4.33 

4.33 

4.33 

4.35 

4.33 

4.33 

4.33 

4.33 

4.33 

10.107190 

60 

1 

.789504 

.896433 

. 893070 

.106930 

59 

2 

.789665 

.896335 

.893331 

.106669 

58 

3 

.789827 

.896236 

.893591 

.106409 

57 

4 

.789988 

.896137 

.893851 

.106149 

56 

5 

.790149 

.896038 

.894111 

.105889 

55 

6 

.790310 

.895939 

.894372 

.105628 

54 

7 

.790471 

.895840 

.894632 

.105368 

53 

8 

.790632 

.895741 

.894892 

.105108 

52 

9 

10 

.790793 

.790954 

.895641 

.895542 

.895152 

.895412 

.104848 

.104588 

51 

50 

11 

9.791115 

2.67 

2.68 

2.67 

2.68 
2.67 
2.67 
2.67 
2.67 
2.67 
2.65 

9.895443 

1.67 
1.65  j 
1.65  j 
1.67.  ; 
1.67 
1.65 
1.67 
1.67 

9.895672 

4.33 

4.33 

4.33 

4.33 

4.32 

4.33 
4.33 
4.33 

4.32 

4.33 

10.104328 

49 

12 

.791275 

.895343 

.895932 

.104068 

48 

13 

.791436 

.895244 

.896192 

.103808 

47 

14 

.791596 

.895145 

.896452 

.103548 

46 

15 

.791757 

.895045 

.896712 

. 103288 

45 

16 

.791917 

.894945 

.896971 

.103029 

44 

17 

.792077 

.894846 

.897231 

.102769 

43 

18 

.792237 

.894746 

.897491 

.102509 

42 

19 

.792397 

.894646 

.897751 

.102249 

41 

20 

.792557 

.894546 

1.67  i 

.898010 

.101990 

40 

21 

9.792716 

2.67 

2.65 

2.67 

2.65 

2.67 

2.65 

2.65 

2.65 

2.65 

2.63 

9.894446 

1.67 
1.67 
1.67  i 
1.67 
1.67 

1.67 

1.68  1 

1.67 

1.68 
1.67 

9.898270 

4.33 

4.32 

4.33 
4 . 32- 
4.33 

4.32 

4.33 
4.32 
4.32 
4.32 

10.101730 

39 

22 

.792876 

.894346 

.898530 

.101470 

38 

23 

.793035 

.894246 

.898789 

.101211 

37 

24 

.793195 

.894146 

.899049 

.100951 

36 

25 

.793354 

.894046 

.899308 

.100692 

35 

26 

.793514 

.893946 

.899568 

.100432 

34 

27 

.793673 

.893846 

.899827 

.100173 

33' 

28 

.793832 

..893745 

.900087 

.099913 

32 

29 

.793991 

.893645 

.900346 

.099654 

'31 

30 

.794150 

.893544 

.900605 

.099395 

30 

31 

9.794308 

2.65 

2.65 

2.63 

2.63 

2.65 

2.63 

2.63 

2.63 

2.63 

2.63 

9.893444 

1.68 

1.67  ! 

1.68  1 
1.68 
1.(8  1 
1.68 

1.67  I 

1.68 
1.70 
1.68 

9.900864 

4.33 
4 .‘32 
4.32 
4.32 

4.32 

4.33 
4.32 
4.32 
4.32 
4.32 

10.099136 

29 

32 

.794467 

.893343 

.901124 

.098876 

28 

33 

.794626 

.893243 

.901383 

.098617 

27 

34 

.794784 

.893142 

.901642 

.098358 

26 

35 

.794942 

.893041 

.901901 

.098099 

25 

36 

.795101 

.892940 

.902160 

.097840 

24 

37 

. 795259 

.892839 

.902420 

.097580 

23 

38 

.795417 

.892739 

.902679 

.097321 

22 

39 

.795575 

.892638 

.902938 

.097062 

21 

40 

. 795733 

.892536 

.903197 

.096803 

20 

41 

9.795891 

2.63 

2.62 

2.63 

2.62 

2.63 

2.62 

2.62 

2.62 

2.62 

2.62 

9.892435 

1.68 

1.68 

1.68 

1.70 

1.68 

1.70 

1.68 

1.70 

1.68 

1.70 

9.903456 

4.30 

4.32 

4.32 

4.32 

4.32 

4.30 

4.32 

4.32 

4.32 

4.30 

10.096544 

19 

42 

.796049 

.892334 

.903714 

.096286 

18 

43 

.796206 

.892233 

.903973 

- .096027 

17 

44 

.796364 

.892132 

.904232 

.095768 

16 

45 

.796521 

.892030 

.904491 

.095509 

15 

46 

i .796679 

.891929 

. 904750 

.095250 

14 

47 

1 .796836 

.891827 

.905008 

.094992 

13 

48 

.796993 

.891726 

.905267 

.094733 

12 

49 

.797150 

.891624 

.905526 

.004474 

11 

50 

.797307 

.891523 

.905785 

.094215 

10 

51 

9.797464 

2.62 

2.60 

2.62 

2.62 

2.60 

2.60 

2.62 

2.60 

2.60 

9.891421 

1.70 

1.70 

1.70 

1.70 

1.70 

1.70 

1.70 

1.70 

1.70 

9.906043 

4.32 

4.30 

4.32 

4.30 

4.32 

4.30 

4.32 

4.30 

4.30 

10.093957 

9 

52 

.797621 

.891319 

.906302 

.093(598 

8 

53 

.7977i7 

.891217 

.906560 

.093440 

7 

54 

.797934 

.891115 

.906819 

.093181 

6 

55 

• .798091 

.891013 

.907077 

.092923 

5 

56 

. 798247 

.890911 

.907336 

.092664 

4 

57 

.798403 

.890809 

.907594 

.092406 

3 

58 

.798560 

.890707 

.907853 

.092147 

2 

59 

.798716 

.890605 

.908111 

.091889 

1 

60 

9.798872 

9.890503 

9.908369 

10.091631 

0 

s 

Cosine. 

D.  1". 

Sine.  1 

D.  r.  i 

i Cotang. 

d.  r. 

Tang. 

' 

128' 


23.6 


6V 


39< 


COSINES,  TANGENTS,  AND  COTANGENTS. 


140* 


' 

Sine. 

d.  r. 

Cosine. 

D.  1". 

Tang.' 

D.  r. 

Cotang. 

/ 

0 

1 

2. 

3* 

4 

5 

6 

8 

9 

10 

11 

12 

13 

14 

15 

16 

17 

18 

19 

20 

21 

22 

23 

24 

25 

26 

27 

28 

29 

30 

31 

32 

33 

34 

35 

36 

37 

38 

39 

40 

41 

42 

43 

44 

45 

46 

47 

48 

49 

50 

51 

52 

53 

54 

55 

56 

57 

58 

59 

60 

9.798872 

.799028 

.799184 

.799339 

.799495 

.799651 

.799806 

.799962 

.800117 

.800272 

.800427 

9.800582 

.800737 

.800892 

.801047 

.801201 

.801356 

.801511 

.801665 

.801819 

.801973 

9.802128 

.802282 

.802436 

.802589 

.802743 

.802897 

.803050 

.803204 

.803357 

.803511 

9.803664 

.803817 

.803970 

.804123 

.804276 

.804428 

.804581 

.804734 

.804886 

.805039 

9.805191 
.805343 
.805495 
.805647 
.805799 
.805951 
.806103 
.806254 
.806406 
. 806557 

9.806709 
.806860 
.807011 
.807163 
1 .807314 

. 807465 
.807615 
.807766 
! .807917 

9.808067 

2.60 

2.60 

2.58 

2.60 

2.60 

2.58 

2.60 

2.58 

2.58 

2.58 

2.58 

2.58 

2.58 

2.58 

2.57 

2.58 
2.58 
2.57 
2.57 

2.57 

2.58 

2.57 

2.57 

2.55 

2.57 

2.57 

2.55 

2.57 

2.55 

2.57 

2.55 

2.55 

2.55 

2.55 

2.55  ' 

2.55 

2.55 

2.55 

2.53 

2.55 

2.53 

2.53 

2.53 

2.53 

2.53 

2.53 

2.53 

2.52 

2.53 
2/52 
2.53 

-2.52 

2.52'' 

2.53 

2.52 

2.52 

2.50 

2.52 

2.52 

2.50 

9.890503 
.890400 
.890298 
.890195 
.890093 
.889990 
.889888 
. 889785 
.889682 
.889579 
.889477 

9.889374 

.889271 

.889168 

.889064 

.888961 

.888858 

.888755 

.888651 

.888548 

.888444 

9.888341 
.888237 
.888134 
.888030 
.887926 
.887822 
.887718 
.887614 
• .887510 

.887406 

9.887302 

.887198 

.887093 

.886989 

.886885 

.886780 

.886676 

.886571 

.886466 

.886362 

9.886257 

.886152 

.886047 

.885942 

.885837 

.885732 

.885627 

.885522 

.885416 

.885311 

9.885205 
.885100 
.884994 
■ .884889 
.884783 
.884677 
.884572 
.884466 
.884360 
9.884254 

1.72 

1.70 

1.72 

1.70 

1.72 

1.70 

1.72 

1.72 

1.72 

1.70 

1.72 

1.72 

1.72 

1.73 
1.72 
1.72 

1.72 

1.73 

1.72 

1.73 

1.72 

1.73 

1.72 

1.73 
1.73 
1.73 
1.73 
1.73 
1.73 
1.73 
1.73 

1.73 

1.75 

1.73 

1.73 

1.75  , 

1.73 

1.75 

1.75 

1.73 

1.75 

1.75 

1.75 

1.75 

1.75 

1.75 

1.75 

1.75 

1.77 

1.75 

1.77 

1.75 
1 1.77 
1.75 
1.77 
1.77 

r 1.75 

1.77 
1 1.77 
1.77 

9.908369 

.908628 

.908.886 

.909144 

.909402 

.909660 

.909918 

.910177 

.910435 

.910693 

.910951 

9.911209 

.911467 

.911725 

.911982 

.912240 

.912498 

.912756 

.913014 

.913271 

.913529 

9.913787 

.914044 

.914302 

.914560 

.914817 

.915075 

.915332 

.915590 

.915847 

.916104 

9.916362 

.916619 

.916877 

.917134 

.917391 

.917648 

.917906 

.918163 

.918420 

.918677 

9.918934 

.919191 

.919448 

.919705 

.919962 

.920219 

.920476 

.920733 

.920990 

.921247 

9.921503 

.921760 

.922017 

.922274 

.922530 

.922787 

.923044 

.923300 

.923557 

9.923814 

4.32 

4.30 

4.30 

4.30 

4.30 

4.30 
4.32 

4.30 

4.30 
4.30 
4.30 

4.30 

4.30 

4.28 

4.30 

4.30 

4.30 

4.30 

4.28 

4.30 

4.30 

4.28 

4.30 

4.30 

4.28 

4.30 

4.28 

4.30 

4.28 

4.28 

4.30 

4.28 

4.30 

4.28 

4.28 

4.28 

4.30 

4.28 

4.28 

4.28 

4.28 

4.28 

4.28 

4.28 

4.28 

4.28 

4.28 

4.28 

4.28 

4.28 

4.27 

4.28 
4.28 
4.28 

4.27 

4.28 
4.28 

4.27  • 

4.28 
4.28 

10.091631 
.091  372 
.091114 
.190856 
.090598 
.090340 
.090082 
.089823 
.089565 
.089307 
.089049 

10.088791 

.088533 

.088275 

.088018 

.087760 

.087502 

.087244 

.086986 

.086729 

.036471 

10.086213 

.085956 

.085698 

.085440 

.085183 

.084925 

.084668 

.084410 

.084153 

.083896 

10.083638 

.083381 

.083123 

.082866 

.082609 

.082352 

.082094 

.081837 

.081580 

.081323 

10.081066 
.080809 
.080552 
.080295 
.080038 
.079781 
.079524 
.079267 
.079010 
. 078753 

10.078497 

.078240 

.077983 

.077726 

.077470 

.077213 

.076956 

.076700 

.076443 

10.076186 

60 

59 

58 

57 

56 

55 

54 

53 

52 

51 

50 

49 

48 

47 

46 

45 

44 

43 

42 

41 

40 

39 

38 

37 

36 

35 

34 

33 

32 

31 

30 

29 

28 

27 

26 

27 

24 

23 

22 

21 

20 

19 

18 

17 

16 

15 

14 

13 

12 

11 

10 

9 

8 

7 

6 

5 

4 

3 

2 

1 

0 

' 1 Cosine. 

d.  r. 

i Sine. 

1 D.  1".  I 

! Cotang. 

D.  1\ 

Tang. 

' 

129° 


237 


50< 


40< 


TABLE  XII.  LOGARITHM  IC  SINES, 


139‘ 


Sine. 


•0 

1 

2 

3 

4 

0 

7 

8 
9 

10 

11 

12 

13 

14 

15 

16 

17 

18 

19 

20 

21 

22 

23 

24 

25 

26 

27 

28 

29 

30 

31 

32 

33 

34 

35 

36 

37 

38 

39 

40 

41 

42 

43 

44 

45 

46 

47 

48 

49 

50 

51 

52 

53 

54 

55 

56 

57 

58 

59 

60 


Cosine. 


D.  1". 


2.52 

2.50 

2.52 

2.50 

2.50 

2.50 

2.50 

2.50 

2.50 

2.50 

2.48 

2.50 

2.48 

2.50 

2.48 

2.48 

2.48 

2.48 

2.48 

2.48 

2.48 

2.47 

2.48 

2.47 

2.48 
2.47 
2.47 
2.47 
2.47 
2.47 
2.47 

2.47 

2.47 

2.45 

2.47 

2.45 

2.47 

2.45 

2.45 

2.45 

2.45 

2.45 

2.45 

2.45 

2.43 

2.45 

2.43 

2.45 

2.43 

2.43 

2.45 

2.43 

2.43 

2.42 

2.43 
2.43 
2.43 

2.42 

2.43 
2.42 

D.  1". 


Cosine.  D.  1". 


9.884254 

.884148 

.884042 

.883936 

.883829 

.883723 

.883617 

.883510 

.883404 

.883297 

.883191 

9.883084 

.882977 

.882871 

.882764 

.882657 

.882550 

.882443 

.882336 

.882229 

.882121 

9.882014 

.881907 

.881799 

.881692 

.881584 

.881477 

.881369 

.881261 

.881153 

.881046 

9.880938 

.880830 

.880722 

.880613 

.880505 

.880397 


.880180 

.880072 

.879963 

9.879855 

.879746 

.879637 

.879529 

.879420 

.879311 

.879202 

.879093 

.878984 

.878875 

9.878766 

.878656 

.878547 

.878438 

.878328 

.878219 

.878109 

.877999 

.877890 

9.877780 


1.77 

1.77 

1.77 

1.78 
1.77 

1.77 

1.78 

1.77 

1.78 

1.77 

1.78 

1.78 

1.77 

1.78 
1.78 
1.78 
1.78 
1.78 
1.78 
1.80 
1.78 

1.78 

1.80 

1.78 

1.80 

1.78 

1.80 

1.80 

1.80 

1.78 

1.80 

1.80 

1.80 

1.82 

1.80 

1.80 

1.80 

1.82 

1.80 

1.82 

1.80 

1.82 

1.82 

1.80 

1.82 

1.82 

1.82 

1.82 

1.82 

1.82 

1.82 

1.83 

1.82 

1.82 

1.83 

1.82 

1.83 

1.83 

1.82 

1.83 


Sine.  I).  1". 


Tang. 

D.  1". 

Cotang. 

' 

9.923814 

4.27 

4.28 

4.27 

4.28 
4.27 

4.27 

4.28 

4.27 

4.28 
4.27 
4.27 

10.076186 

60 

.924070 

.075930 

59 

.924327 

.075673 

58 

.924583 

.075417 

57 

.924840 

.075160 

56 

.925096 

.074904 

55 

.925352 

.074648 

54 

.925609 

.074391 

53 

.925865 

.074135 

52 

.926122 

.073878 

51 

.926378 

.073622 

50 

9.926634 

4.27 

4.28 
4.27 
4.27 
4.27 
4.27 

4.27 

4.28 
4.27 
4.27 

10.073366 

49 

.926890 

.073110 

48 

.927147 

.072853 

47 

.927403 

.072597 

46 

.927659 

.072341 

45 

.927915 

.072085 

44 

.928171 

.071^29 

43 

.928427 

.071573 

42 

.928684 

.071316 

41 

.928940 

.071060 

40 

9.929196 

4.27 

4.27 

4.27 

4.27 

4.25 

4.27 

4.27 

4.27 

4.27 

4.27 

10.070804 

39 

.929452 

.070548 

38 

.929708 

.070292 

37 

.929964 

.070036 

36 

.930220 

.069780 

35 

.930475 

.069525 

34 

.930731 

.069269 

33 

.930987 

.069013 

32 

.931243 

.068757 

31 

.931499 

. 068501 

30 

9.931755 

4.25 

4.27 

4.27 

4.27 

4.25 

4.27 

4.27 

4.25 

4.27 

4.25 

10.068245 

29 

.932010 

.067990 

! 28 

.932266 

.067734 

27 

.932522 

.067478 

26 

.932778 

.067222 

25 

.933033 

.066967 

24 

.933289 

.066711 

23 

.933545 

. 066455 

22 

.933800 

.066200 

21 

.934056 

.065944 

20 

9.934311 

4.27 

4.25 

4.27 

4.25 

4.27 

4.25 

4.27 

4.25 

4.27 

4.25 

10.065689 

19 

.934567 

.065433 

18 

.934822 

.065178 

17 

.935078 

.064922 

16 

.935333 

.064667 

15 

.935589 

.064411 

14 

.935844 

.064156 

13 

.936100 

.063900 

12 

.936355 

.063645 

11 

.936611 

.063389 

10 

9.936866 

a on 

10.063134 

9 

.937121 

4.27 

4.25 

4.25 

4.25 

4.27 

4.25 

4.25 

4.25 

.062879 

8 

.937377 

.062628 

7 

.937632 

.062368 

6 

.937887 

.938142 

.062113 
.061858  | 

5 

4 

.938398 

.061602 

3 

.938653 

.061347 

2 

.938908 

.061092 

1 

9.939103 

10.060837 

0 

1 Cotang. 

d.  r. 

Tang. 

' 

41° 

COS 

/ 

Sine.  1 

0 

9.816943 

1 

.817088 

2 

.817233 

3 

.817379 

4 

.817524 

5 

.817668 

6 

.817813 

7 

.817958 

8 

.818103 

9 

.818247 

10 

.818392 

11 

9.818536 

12 

.818681 

13 

.818825 

14 

.818969 

15 

.819113 

16 

.819257 

17 

.819401 

18 

.819545 

19 

.819689 

20 

.819832 

21 

9.819976 

22 

.820120 

23 

.820263 

24 

.820406 

25 

. 820550 

26 

.820693 

27 

.820836 

28 

.820979 

29 

.821122 

30 

.821265 

31 

9.821407 

32 

.821550 

33 

.821693 

34 

.821835 

35 

.821977 

36 

.822120 

37 

.822262 

38 

.822404 

39 

.822546 

' 40 

.822683 

41 

9.822830 

42 

.822972 

43 

.823114 

44 

.823255 

45 

.823397 

46 

.823539 

47 

.823680 

48 

.823821 

49 

.823963 

50 

.824104 

51 

9.824245 

52 

.824386 

53 

.824527 

54 

.824668 

55 

.824808 

56 

.824949 

57 

.825090 

58 

! .825230 

59 

| .825371 

60 

9.825511 

' 1 Cosine. 

TANGENTS,  AND  COTANGENTS. 


138° 


2.42 

2.42 

2.43 
2.42 
2.40 
2.42 
2.42 
2.42 
2.40 
2.42 
2.40 

2.42 

2.40 

2.40 

2.40 

2.40 

2.40 

2.40 

2.40 

2.38 

2.40 

2.40 

2.38 

2.33 

2.40 

2.38 

2.38 

2.38 

2.38 

2.38 

2.37 

2.38 
2.38 
2.37 

2.37 

2.38 
2.37 
2.37 
2.37 
2.37 
2.37 

2.37 

2.37 

2.35 

£.37 

2.37 

2.35 

2.35 

2.37 

2.35 

2.35 

2.35 

2.35 

2.35 

2.33 

2.35 

2.35 

2.33 

2.35 

2.33 

D.  r. 


Cosine. 


D.  1". 


9.877780 
.877670 
. 877560 
.877450 
.877340 
.877230 
.877120 
.877010 
.876899 
.876789 
.876678 

9.876568 

.876457 

.876347 

.876236 

.876125 

.876014 

.875904 

.875793 

.875682 

.875571 

9.875459 

.875348 

.875237 

.875126 

.875014 

.874903 

.874791 

.874680 

.874568 

.874456 

9.874344 

.874232 

.874121 

.874009 

.873896 

.873784 

.873672 

.873560 

.873448 

.873335 

9.873223 

.873110 

.872998 

.872885 

.872772 

.872659 

.872547 

.872434 

.872321 

.872208 

9.872095 

.871981 

.871868 

.871755 

.871641 

.871528 

.871414 

.871301 

.871187 

9.871073 

Sine. 


134° 


1.83 

1.83 

1.83 

1.83 

1.83 

1.83 

1.83 

1.85 

1.83 

1.85 

1.83 

1.85 

1.83 

1.85 

1.85 

1.85 

1.83 

1.85 

1.85 

1.85 

1.87 

1.85 

1.85 

1.85 

1.87 

1.85 

1.87 

1.85 

1.87 

1.87 

1.87 

1.87 

1.85 

1.87 

1.88 
1.87 
1.87 
1.87 

1.87 

1.88 

1.87 

1.88 

1.87 

1.88 
1.88 
1.88 

1.87 

1.88 
1.88 
1.88 
1.88 

1.90 

1.88 

1.88 

1.90 

1.88 

1.90 

1.88 

1.90 

1.90 

D.  1". 


239 


1 Tang. 

d.  r. 

Cotang. 

/ 

9.939163 

4.25 

4.25 

4.25 

4.25 

4.27 

4.25 

4.25 

4.25 

4.25 

4.23 

4.25 

10.060837 

60 

.939418 

.060582 

59 

.939673 

.060327 

58 

.939928 

.060072 

57 

.940183 

.059817 

56 

.940439 

.059561 

55 

.940694 

.059306 

54 

.940949 

.059051 

53 

.941204 

.058796 

52 

.941459 

.058541 

51 

.941713 

.058287 

50 

9.941968 

4.25 

4.25 

4.25 

4.25 

4.25 

4.25 

4.23 

4.25 

4.25 

4.25 

10.058032 

49 

.942223 

.057777 

48 

.942478 

.057522 

47 

.942733 

.057267 

46 

.942988 

.057012 

45 

.943243 

.056757 

44 

.943498 

.056502 

43 

.943752 

.056248 

42 

.944007 

.055993 

41 

.944262 

.055738 

40 

9.944517 

4.23 

4.25 

4.25 

4.23 

4.25 

4.25 

4.23 

4.25 

4.23 

4.25 

10.055483 

39 

.944771 

.055229 

38 

.945026 

.054974 

37 

.945281 

.054719 

36 

.945535 

.054465 

35 

.945790 

.054210 

34 

.946045 

.053955 

33 

.946299 

.053701 

32 

.946554 

.053446 

31 

.946808 

.053192 

30 

9.947063 

4.25 

4.23 

4.25 

4.23 

4.23 

4.25 

4.23 

4.25 

4.23 

4.25 

10.052937 

29 

.947318 

.052682 

28 

.947572 

.052428 

27 

.947827 

.052173 

26 

.948081 

.051919 

25 

.948335 

.051665 

24 

.948590 

.051410 

23 

.948844 

.051156 

22 

.949099 

.050901 

21 

.949353 

.050647 

20 

9.949608 

4.23 

4.23 

4.25 

4.23 

4.23 

4.23 

4.25 

4.23 

4.23 

4.23 

10.050392 

19 

.949862 

.050138 

18 

.950116 

.049884 

17 

.950371 

.049629 

16 

.950625 

.049375 

15 

.950879 

.049121 

14 

.951133 

.048867 

13 

.951388 

.048612 

12 

.951642 

.048358 

11 

.951896 

.048104 

10 

9.952150 

4.25 

4.23 

4.23 

4.23 

4.23 

4.23 

4.23 

4.23 

4.23 

10.047850 

9 

.952405 

.047595 

8 

.952659 

.047341 

7 

.952913 

.047087 

6 

.953167 

.046833 

5 

. 953421 

.046579 

4 

.953675 

.046325 

3 

.953929 

.046071 

2 

.954183 

.045817 

1 

9.954437 

10.045563 

0 

! Cotang. 

1 D.  1". 

1 Tang. 

I ' 

48° 


42° 


TABLE  XII.  LOGARITHMIC  SINES, 


137 


/ 

Sine. 

D.  1\ 

Cosine. 

I),  r. 

Tang. 

D.  r. 

Cotang. 

/ 

0 

9.825511 

2.33 
2. ,33 
2.33 
2.33 
2.33 
2.33 
2.33 
2.33 

2.32 

2.33 
2.32 

9.871073 

1.88 

1.90 

1.90 

1.90 

1.90 

1.90 

1.90 

1.92 

1.90 

1.90 

1.92 

1 9.954437 

4.23 

4.25 

4.23 

4.23 

4.23 

4.22 

4.23 
4.23 
4.23 
4.23 
4.23 

10.045563 

60 

1 

.825651 

.870960 

.954691 

.045309 

59 

2 

.825791 

.870846 

.954946 

.045054 

58 

3 

.825931 

.870732 

.955200 

.044800 

57 

4 

.826071 

.870618 

.955454 

.044546 

56 

5 

.826211 

.870504 

.955708 

.044292 

55 

C 

.826351 

.870390 

.955961 

.044039 

54 

7 

.826491 

.870276 

.956215 

.043785 

53 

8 

.826631 

.870161 

.956469 

.043531 

52 

9 

.826770 

.870047 

.956723 

.043277 

51 

10 

.826910 

.869933 

.956977 

.043023 

50 

11 

9.827049 

2.33 

2.32 

2.32 

2.32 

2.32 

2.32 

2.32 

.2.32 

2.32 

2.30 

9.869818 

1.90 

1.92 

1.92 

1.90 

1.92 

1.92 

1.92 

1.92 

1.92 

1.92 

9.957231 

4.23 

4.23 

4.23 

4.23 

4.22 

4.23 
4.23 
4.23 
4.23 
4.22 

10.042769 

49 

12 

.827189 

.869704 

.957485 

.042515 

48 

13 

.827328 

.869589 

.957739 

.042261 

47 

14 

.827467 

.869474 

.957993 

.042007 

46 

15 

.827606 

.869360 

.958247 

.041753 

45 

16 

.827745 

.869245 

.958500 

.041500 

44 

17 

.827884 

.869130 

.958754 

.041246 

4 3 

18 

.828023 

.869015 

.959008 

.040992 

42 

19 

.828162 

.868900 

.959262 

.040718 

41 

20 

.828301 

.868785 

.959516 

.040484 

40 

21 

9.828439 

2.32 
‘ 2.30 
2.32 
2.30 
2.30 
2.30 
2.30 
2.30 
2.30 
2.30 

9.868670 

1.92 

1.92 

1.93 

1.92 

1.93 

1.92 

1.93 

1.92 

1.93 
1.93 

9.959769 

4.23 

4.23 

4.22 

4.23 
4.23 
4.23 

4.22 

4.23 

4.22 

4.23 

10.040231 

39 

22 

.828578 

.868555 

.960023 

.039977 

38 

23 

.828716 

.868440 

.960277 

.039723 

37 

24 

,828855 

.868324 

.960530 

.039470 

36 

25 

.828993 

.868209 

.960784 

.039216 

35 

26 

.829131 

.868093 

.961038 

.038962 

34 

27 

.829269 

867978 

.961292 

.038708 

33 

28 

.829407 

.867862 

. 961545 

.038455 

32 

29 

.829545 

.867747 

.961799 

.038201 

31 

30 

.829683 

.867631 

.962052 

.037948 

30 

31 

9.829821 

2.30 

2.30 

2.28 

2.30 

2.28 

2.28 

2.30 

2.28 

2.28 

2.28 

9.867515 

1.93 

1.93 

1.93 

1.93 

1.93 

1.93 

1.93 

1.95 

1.93 

1.95 

9.962306 

4.23 

4.22 

4.23 

4.22 

4.23 
4.23 

4.22 

4.23 

4.22 

4.23 

10.037694 

29 

32 

33 

.829959 

.830097 

.867399 

.867283 

.962560' 

.962813 

.037440 

.037187 

28 

27 

34 

.830234 

.867167 

| .963067 

.036933 

26 

35 

.830372 

.867051 

.963320 

.036680 

25 

36 

.830509 

.866935 

.963574 

.036426 

24 

37 

.830646 

.866819 

.963828 

.036172 

23 

38 

.830784 

.866703 

.964081 

.035919 

22 

39 

.830921 

.866586 

.964335 

.035665 

21 

40 

.831058 

.866470 

.964588 

.035412 

20  . 

41 

9.831195 

2.28 

2.28 

2.28 

2.27 

2.28 

2.27 

2.28 

2.27 

2.28 
2.27 

9.866353 

1.93 

1.95 

| 1.93  I 
1.95 
1.95 

9.964842 

4.22 

4.23 
4.22 
4.22 

A OQ 

10.035158 

19 

42 

.831332 

.866237 

.965095 

.034905 

18 

43 

.831469 

.866120 

.965349 

.034651 

17 

44 

.831606 

.866004 

. 965602 

.034398 

16 

45 

.831742 

.865887 

.965855 

.034145 

15 

46 

.831879 

.865770 

! .966109 

4.22 

4.23 

4.22 

4.23 
4.22 

.033891 

14 

47 

.832015 

.865653 

J . JO  | 

! 1.95 
1.95  | 
1.95  | 
1.95  j 

j .966362 

.033638 

13 

48 

.832152 

.865536 

| .966616 

.033384 

12 

49 

.832288 

.865419 

! .966869 

.033131 

11 

50 

.832425 

.865302 

.967123 

.032877 

10 

51 

9-832561 

2.27 

2.27 

2.27 

2.27 

2.27 

2.27 

2.25 

2.27 

2.25 

9.865185 

1.95 

1.97 

1.95 

1.95 

1.97 

1.95 

1.97 

1.97 

1.97 

9.967376 

4.22 

4.23 
4.22 

4.22 

4.23 
4.22 

4.22 

4.23 

10.032624 

9 

52 

.832697 

.865068 

.967629 

.032371 

8 

53 

.832833 

.864950 

.967883 

.032117 

7 

54 

832969 

.864833 

| .968136 

.031864 

6 

: 55 

.833105 

.864716 

.968389 

.031611 

5 

1 56 

.833241 

.864598 

.968643 

.031357 

4 

! 57 

.833377 

.864481 

.968896 

.031104 

3 

1 58 

.833512 

.864363  j 

j .969149 

.030851 

0 

; 59 

.833648 

.864245  I 

.969403 

.030597 

1 

j 60 

9.833783 

9.864127 

9.969656 

4.22 

10.030344 

0 

! ' 

Cosine. 

D.  1*. 

Sine, 

D.  1", 

Cotang. 

D.  1\ 

Tang. 

47* 


132‘ 


240 


COSINES,  TANGENTS,  AND  C(^T|A(l^^lf v(t^. 


t 

Sine. 

tl 

Cosine. 

D.  1". 

1 1 1 ! 

| Tang. 

D.  1". 

Cotang. 

i 

0 

9.833783 

2.27 

2.25 

2.25 

2.27 

2.25 

2.25 

2.25 

2.25 

2.23 

2.25 

2.25 

9.864127 

1.95 

1/97 

1.97 

1.97 

1.97 

1.98 
1.97 

1.97 

1.98 

1.97 

1.98 

9.969656 

4.22 

4.22 

4.23 

4.2 1 

4.22 

10.030344 

60 

1 

.833919 

.864010 

.969909 

.030091 

59 

2 

.834054 

.863892 

.970162 

.029838 

58 

3 

.834189 

.863774 

.970416 

.029584 

57 

4 

. 834325 

. 863656 

.970669 

.029331 

56 

5 

.834460 

.863538 

.970922 

.029078 

55 

6 

.834595 

.863419 

.971175 

A 23 
4.22 
4.22 
4.22 
4.22 

.028825 

54 

7 

.834730 

.863301 

.971429 

.028571 

53 

8 

.834865 

.863183 

.971682 

.028318 

52 

9 

.834999 

.863064 

.971935 

.028065 

| 51 

10 

.835134 

.862946 

.972188 

.027812 

; 50 

11 

9.835269 

2.23 

2.25 

2.23 

2.25 

2.23 

2.23 

2.23 

2.23 

2.23 

2.23 

9.862827 

1.97 

1.98 
1.98 

1.97 

1.98 
1.98 
1.98 
1.98 
1.98 
2.00 

9.972441 

4.23 

4.22 

4.22 

4.22 

4.22 

4.22 

4.22 

4.22 

4.23 
4.22 

10.027559 

49 

12 

.835403 

.862709 

.972695 

.027305 

48 

13 

.835538 

.862590 

.972948 

.027052 

i 47 

14 

.835672 

.862471 

.973201 

.026799 

46 

15 

.835807 

.862353 

.973454 

.026546 

45 

16 

.835941 

.862234 

. 973707 

.026293 

44 

17 

18 

. 836075 
.836209 

.862115 

.861996 

.973960 

.974213 

.026040 

.025787 

43 

42 

19 

.836343 

.861877 

.974466 

.025534 

41 

20 

.836477 

.861758 

.974720 

.025280 

40 

21 

9.836611 

2.23 

2.22 

2.23 

2.23 

2.22 

2.22 

2.2£ 

2.22 

2.22 

2.22 

9.861638 

1.98 

1.98 

2.00 

1.98 

2.00 

9.974973 

4.22 

10.025027 

39 

22 

.836745 

.861519 

.975226 

.024774 

38 

23 

.836878 

.861400 

.975479 

4*22 

4.22 

4.22 

4.22 

4.22 

4.22 

4.22 

4.22 

.024521 

37 

24 

.837012 

.861280 

.975732 

.024268 

36 

25 

.837146 

.861161 

.975985 

.024015 

35 

26 

.837279 

.861041 

. 976238 

.023762 

34 

27 

.837412 

.860922 

1.98 

2.00 

2.00 

2.00 

2.00 

.976491 

■ .023509 

33 

28 

.837546 

.860802 

.976744 

.023256 

32 

29 

.837679 

.860682 

.976997 

.023003 

31 

30 

.837812 

.860562 

.977250 

.022750 

30 

31 

9.837945 

2.22 
2.22 
2.22 
2.22  . 
2.22 
2.20 
2.22 
2.20 
2.22 
2.20 

9.860442 

2.00 

2.90 

2.00 

2.00 

2.00 

2.02 

2.00 

2.02 

2.00 

2.02 

9.977503 

4.22 

4.22 

.4.22 

4.22 

4.22 

4.22 

4.22 

4.22 

4.22 

4.22 

10.022497 

29 

32 

.838078 

.860322 

.977756 

.022244 

28 

33 

.838211 

.860202 

.978009 

.021991 

27 

34 

.838344 

.860082 

.978262 

.021738 

1 26 

35 

.838477 

.859962 

. .978515 

.021485 

i 25 

36 

.838610 

.859842 

.978768 

.021232 

24 

37 

.838742 

.859721 

.979021 

.020979 

; 23 

3^ 

.838875 

.859601 

.979274 

.020726 

I 22 

39 

.839007 

.859480 

.979527 

.020473 

1 21 

• 40 

.839140 

.859360 

.979780 

.020220 

i 20 

41 

9.839272 

2.20 

2.20 

2.20 

2.20 

2.20 

2.20 

2.20 

2.20 

2.18 

2.20 

9.859239 

2.00 

2.02 

2.02 

2.02 

2.02 

2.02 

2.02 

2.02 

2.02 

2.03 

9.980033 

4.22 

4120 

4.22 

4.22 

4.22 

4.22 

4.22 

4.22 

4.22 

4.22 

10.019967 

! 19 

42 

.839404 

.859119 

.980286 

.019714 

18 

43 

.839536 

.858998 

.980538 

.019462 

! 17 

44 

.839668 

.858877 

.980791 

.019209 

I 16 

45 

.839800 

.858756 

.981044 

.018956 

15 

46 

.839932 

.858635 

.981297 

.018703 

14 

47 

.840064 

.858514 

.981550 

.018450 

! 13 

48 

.840196 

.858393 

.981803 

.018197 

12 

49 

.840328 

.858272 

.982056 

.017944 

11 

50 

.840459 

.858151 

..982309 

.017691 

10 

51 

9.840591 

. 2.18 
2.20 
2.18 
2.18 
2.18 
2.18 
2.18 
2.18 
2.18 

9.858029 

2.02 

9.982562 

4.20 

10.017438 

9 

52 

.840722 

.857908 

.982814 

.017186 

8 

53 

.840854 

.857786 

2 . 03 

.983067 

4.22 

4.22 

4.22 

4.22 

4.22 

4.22 

4.20 

4.22 

.016933 

7 

54 

.840985 

.857665 

2.02 
2.03 
2.02 
I 2.03 
1 2.03 
1 2.03 
2.03 

.983320 

.016680 

6 

55 

.841116 

.857543 

.983573 

.016427 

5 

56 

.841247 

.857422 

.983826 

.016174 

4 

57 

.841378 

.857300 

.984079 

.015921 

3 

58 

.841509 

.857178 

.984332 

.015668 

2 

59 

.841640 

.857056 

.984584 

.015416 

1 

60 

9.841771 

9.856934 

9.984837 

10.015163 

0 

6 

1 Cosine. 

D.  1". 

1 Sine. 

L D.  r. 

Cotang.  1 

D.  1". 

Tang. 

/ 

44° 


135' 

1 


T^#Ljl£  ixfn.  LOGARITHMIC  SINES, 


! 1 .M  ! 

Sine. 

D.  r. 

j 

Cosine. 

b.  r. 

Tang. 

D.  1'. 

Cotang. 

/ 

0 

9.841771 

2.18 

2.18 

2.17 

2.18 

2.17 

2.18 
2.17 

2.17 

2.18 
2.17 
2.17 

9.856934 

2.03 

2.03 

2.03 

2.03 

2.05 

2.03 

2.05 

2.03 

2.05 

2.03 

2.05 

9.984837 

4.22 

4.22 

4.22 

4.20 

4.22 

4.22 

4.22 

4.22 

4.20 

4.22 

4.22 

10.015163 

60 

1 

.841902 

.856812 

.985090 

.014910 

59 

2 

.842033 

.856690 

.985343 

.014657 

58 

3 

.842163 

. 856568 

.985596 

014404 

57 

4 

.842294 

. 856446 

.985848 

.014152 

56 

5 

.842424 

.856323 

.986101 

.013899 

55 

6 

. 842555 

.856201 

986354 

.013646 

54 

7 

.842685 

.856078 

.986607 

.013393 

53 

8 

.842815 

. 855956 

.986860 

.013140 

52 

9 

.842946 

.855833 

.987112 

.012888 

51 

10 

.843076 

.855711 

.987365 

.012635 

50 

11 

9.843206 

2.17 

2.17 

2.15 

2.17 

2.17 

2.15 

2.17 

2.15 

2.15 

2.17 

9.855588 

2.05 

2.05 

2.05 

2.05 

2.05 

2.05 

2.05 

2.07 

2.05 

2.07 

9.987618 

4.22 

4.20 

4.22 

4.22 

4.22 

4.20 

4.22 

4.22 

4.22 

4.20 

10.012382 

49 

12 

.843336 

.855465 

.987871 

.012129 

48 

13 

.843466 

.855342 

.988123 

.011877 

47 

14 

.843595 

.855219 

.988376 

.011624 

46 

15 

.843725 

.855096 

.988629 

.011371 

45 

16 

.843855 

.854973 

.988882 

.011118 

44 

17 

.843984 

.854850 

.989134 

.010866 

43 

18 

.844114 

.854727 

.989387 

.010613 

42 

19 

.844243 

.854603 

.989640 

.010360 

41 

20 

.844372 

.854480 

.989893 

.010107 

40 

21 

9.844502 

2.15 

2.15 

2.15 

2.15 

2.15 

2.15 

2.15 

2.13 

2.15 

2.13 

9.854356 

© ©bo© © boob 

9.990145 

4.22 

4.22 

4.20 

4.22 

4.22 

4.22 

4.20 

4.22 

4.22 

4.20 

10.009855 

39 

22 

.844631 

.854233 

.990398 

.009602 

38 

23 

.844760 

.854109 

.990651 

.009349 

37 

24 

.844889 

.853986 

.990903 

.009097 

36 

25 

.845018 

.853862 

.991156 

.008844 

35 

26 

.845147 

.853738 

.991409 

.008591 

34 

27 

.845276 

.853614 

.991662 

.008338 

33 

28 

.845405 

.853496 

.991914 

.008086 

32 

29 

.845533 

.853366 

.992167 

.007833 

31 

30 

.845662 

.853242 

.992420 

.007580 

30 

31 

9.845790 

2.15 

2.13 

2.13 

2.15 

2.13 

2.13 

2.13 

2.13 

2.13 

2.12 

9.853118 

2.07 

2.08 

2.07 

2.08 

2.07 

2.08 

2.07 

2.08 
2.08 
2.08 

9.992672 

4.22 

4.22 

4.22 

4.20 

4.22 

4.22 

4.20 

4.22 

4.22 

4.20 

10.007328 

29 

32 

.845919 

.852994 

.992925 

.007075 

28 

33 

.846047 

.852869 

.993178 

.006822 

27 

34 

.846175 

.852745 

.993431 

.006569 

26 

35 

.846304 

.852620 

.993683 

.006317 

25 

36 

.846432 

.852496 

.993936 

.006064 

24 

37 

.846560 

.852371 

.994189 

.005811 

23 

38 

.846688 

.852247 

.994441 

.005559 

22 

39 

.816816 

.852122 

.994694 

.005306 

21 

40 

.846944 

.851997 

.994947 

.005053 

20 

41 

9.847071 

2.13 

2.13 

2.12 

2.13 

2.12 

2.12 

2.13 

2.12 

2.12 

2.12 

9.851872 

2.08 

2.08 

2.08 

2.08 

2.10 

2.08 

2.08 

2.10 

2.08 

2.10 

9.995199 

4.22 

4.22 

4.20 

4.22 

4.22 

4.20 

4.22 

4.22 

4.20 

4.22 

10.004801 

19 

42 

.847199 

.851747 

.995452 

.004548 

18 

43 

.847327 

.851622 

.995705 

.004295 

17 

44 

.847454 

.851497 

.995957 

.004043 

16 

45 

.847582 

.851372 

.996210 

.003790 

15 

46 

.847709 

.851246 

.996463 

.003537 

14 

47 

.847836 

.851121 

.996715 

.003285 

13 

48 

.847964 

.850996 

.996968 

.003032 

12 

49 

.848091 

.850870 

.997221 

.002779 

11 

50 

.848218 

.850745 

.997473 

.002527 

10 

51 

9.848345 

2.12 

2.12 

2.12 

2.10 

2.12 

2.12 

2.10 

2.12 

2.10 

9.850619 

2.10 
2.08 
2.10 
2.10 
2.10 
2.10 
2.10  I 
2.12  i 
2.10 

9.997726 

4.22 

4.20 

4.22 

4.22 

4.20 

4.22 

4.22 

4.20 

4.22 

10.002274 

9 

52 

.848472 

.850493 

.997979 

.002021 

8 

53 

.848599 

.850368 

.998231 

.001769 

7 

54 

.848726 

.850242 

.998484 

.001516 

6 

55 

.848852 

.850116 

.998737 

.001263 

5 

56 

.848979 

.849990 

.998989 

.001011 

4 

57 

.849106 

.849864 

.999242 

.000758 

3 

58 

.849232 

.849738 

.999495 

.000505 

2 

59 

.849359 

.849611 

.999747 

.000253 

1 

60 

9.849485 

9.849485 

10.000000 

10.000000 

0 

' 1 Cosine. 

D.  1". 

Sine. 

D.  1\  I 

Cotang. 

D.  r. 

Tang. 

/ 

134°  45* 


242 


AZIMUTH  BY  ALTITUDE  OF 


?!  iifkll  / 


243 


Art.  41.  Azimuth  by  Altitude  of  Sun. 


The  azimuth  of  a given  line  m\y  be  determined  by  taking 
the  altitude  of  the  sun  with  an  engineers’  transit  having  a 
good  vertical  circle,  and  reading  the  horizontal  angle  between 
the  sun  and  the  line.  The  latitude  of  the  place  must  be 
known  and  a nautical  almanac  must  be  at  hand  for  finding  the 
declination  of  the  sun  at  the  moment  of  observation. 

In  Fig.  59  let  A,  represent  the  center  of  the  celestial  sphere, 
Z the  zenith,  P the  pole,  N the  north  point  of  the  horizon,  S 
the  position  of  the  sun  at  the  moment  of  observation.  Then, 
in  the  spherical  triangle  PZS,  the  angle  Z is  the  azimuth  of 
the  sun,  and  this  is  the  same  as  the  horizontal  angle  NAG.  If 
AB  be  the  line  whose  azimuth  is  to  be  found,  NAB  is  its 
azimuth.  Now  if  the  horizontal  angle  BAG  be  measured,  and 
Z be  computed,  the  azimuth  of  AB  is  known. 

To  find  the  azimuth  of  the  sun  Z,  let  z be  the  complement  of 
the  observed  altitude  GS,  corrected  for  refraction  and  parallax  ; 
let  (p  be  the  latitude  of  the  place,  or  the  arc  NP  ; let  d be  the 
declination  of  the  sun  or  the  arc  QS.  Then  in  the  spherical 
triangle  PZS  three  sides  are  known,  and  hence 


tan  ±Z 


*=•/: 


cos  J(2  + 0 + £)  sin  \(z  -f-  <p  ~ $) 
cos  \{z  — (p  — 8)  sin  \ z — (p  -f-  d)’ 


from  which  the  azimuth  Z can  be  computed. 

In  the  figure  S denotes  the  place  of  the  sun  in  the  summer 
half-year  when  d is  positive, 
and  S'  its  place  in  the  winter 
half-year  when  d is  negative. 

If  the  observation  be  made  in 
the  forenoon,  the  value  of  Z is 
less  than  180  degrees  ; if  it  be 
made  in  the  afternoon,  its  value 
is  greater  than  180  degrees. 

The  transit  having  been  put 
into  thorough  adjustment,  it  is 
set  up  at  A,  the  end  of  the  line 
AB,  whose  azimuth  is  to  be 
found.  The  vernier  of  the 
horizontal  limb  having  been  set  at  0°  00/  the  telescope  is 
pointed  at  B and  the  alidade  unclamped.  The  telescope  is 


Fig.  59. 


244\'  ( jA^lMUTU  BY  ALTITUDE  OE  SUIT. 

then  pointed  upon  the  sun,  the  objective  and  eyepiece  being 
so  focused  that  the  shadow  of  the  cross-wires  and  the  image  of 
the  sun  may  be  plainly  seen  on  a white  piece  of  paper  held 
behind  the  eyepiece.  The  cross- wires  should  be  made  tangent 
to  the  bright  circle  on  its  lower  and  right-hand  sides,  and  the 
horizontal  and  vertical  angles  be  read.  Next,  the  cross- wires 
should  be  made  tangent  on  the  upper  and  left-hand  sides  of 
the  bright  circle,  and  the  angles  be  read  again.  If  the  transit 
has  a full  vertical  circle,  which  is  necessary  for  the  best  work, 
observations  should  be  taken  both  in  the  direct  and  reverse 
position  of  the  telescope. 

The  following  record  of  an  observation  will  illustrate  the 
method  of  making  the  measurements  and  obtaining  the  data 
for  computation.  The  declination  8 for  8:43  am.,  eastern 
standard  time,  of  the  day  of  observation,  is  here  taken  from  a 
nautical  almanac,  but  for  general  purposes  it  may  be  taken 


Time 

Vertical 

Horizontal 

May  19, 

Tel. 

Angle. 

Angle. 

Data  and  Results. 

1897. 

CAS 

BAC 

A.M. 

Wires  tang 

ent  to  lower 

0 = 40°  36'  27" 

and  right 

sides. 

6 at  7 a.m.  = 19°  53'  10" 

55 

8h  40m 

D 

43°  09'  00" 

64°  48'  00" 

- 

5 = 19°  54'  05" 

42 

R 

43  35  30 

65  10  30 

Appar.  Alt.  = 43°  58'  22" 

Parallax...  +06 

Wires  tang 
and  left 

ent  to  upper 
sides. 

Refraction.  - 60 

Altitude  = 43°  57'  28" 

90  00  00 

8 44 

R 

o 

o 

05 

o 

Mi 

rf 

64°  52'  30" 

z = 46°  02'  32" 

46 

D 

44  48  00 

65  15  00 

Z = 98°  08'  08"  • 

65  01  30 

Means  = 

43°  58'  22" 

65°  OR  30" 

NAB  = 33°  06'  38" 

from  the  solar  table  mentioned  on  page  126.  The  mean  ap- 
parent altitude  is  43°  58'  22,"  and  tliis  being  corrected  for 
parallax  and  refraction,  the  zenith  distance  z is  found.  By 
computation  from  the  formula,  the  mean  azimuth  of  the  sun  is 
98°  08'  08,"  and  subtracting  from  this  the  mean  horizontal 
angle  BAG  the  final  azimuth  of  the  line  AB  is  33°  06'  38/' 

The  uncertainty  of  an  azimuth  found  by  this  method  is  two 


MEAN  REFRACTION, 


245 

, . r'  ■ i iiir 

or  three  minutes.  The  best  time  for  observation  is  when  the 
bearing  of  the  sun  is  nearly  east  or  nearly  west,  and  for  any 
precise  work  a mean  result  should  be  determined  by  several 
morning  and  afternoon  observations. 

The  correction  for  parallax  of  the  sun  is  less  than  8". 6,  and 
is  always  added  to  the  apparent  altitude ; for  an  altitude  of  20° 
the  parallax  correction  is  8",  for  40°  it  is  7",  and  for  60°  it  is 
6".  In  precise  computations  the  value  of  the  parallax  cor- 
rection may  be  found  by  multiplying  8". 6 by  the  cosine  of  the 
apparent  altitude  of  the  sun. 


The  correction  for  refraction  is  always  subtracted  from  the 
apparent  altitude,  and  its  value  is  to  be  taken  from  the  follow- 
ing table,  interpolating  when  necessary. 


Table  XIII.  Mean  Refractions. 


Apparent 

Altitude. 

Refraction. 

Apparent 

Altitude. 

Refraction. 

Apparent 

Altitude. 

Refraction. 

Apparent 

Altitude. 

Refraction. 

0° 

34'  54" 

20° 

2'  37" 

40° 

69'' 

60° 

33" 

1 

24  25 

21 

2 29 

41 

66 

61 

32 

2 

18  09 

22 

2 22 

42 

64  - 

62 

31 

3 

14  15 

23 

2 15 

43 

62 

63 

29 

4 

11  39 

24 

2 09 

44 

60 

64 

28 

5 

9 46 

25 

2 03 

45  . 

58 

65 

27 

6 

8 23 

26 

1 58 

46 

56 

66 

26 

7 

7 20 

27 

1 53 

47 

54 

67 

24 

8 

6 30 

28 

1 48 

48 

52 

68 

23 

9 

5 49 

29 

1 44 

49 

50 

69 

22 

10 

5 16 

' 30 

1 40 

50 

48 

70 

21 

11 

4 49 

31 

1 36 

51 

47 

72 

19 

12 

4 25 

32 

1 32 

52 

45 

74 

17 

13 

4 05 

33 

1 29 

53 

43 

76 

15 

14 

3 47 

34 

1 25 

54 

42 

78 

12 

15 

3 32 

35 

1 22 

55 

40 

80 

10 

16 

3 19 

36 

1 19 

56 

39 

82 

8 

17 

3 07 

37 

1 16 

57 

38 

84 

6 

18 

2 56 

38 

1 14 

58 

36 

86 

4 

19 

2 46 

39 

1 11 

59 

35 

88 

2 

20 

2 37 

40 

1 09 

60 

33 

90 

0 

Gothic. 


246 


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